Isqrt (integer square root) of X
From Rosetta Code
You are encouraged to solve this task according to the task description, using any language you may know.
Sometimes a function is needed to find the integer square root of X, where X can be a real non─negative number.
Often X is actually a non─negative integer.
For the purposes of this task, X can be an integer or a real number, but if it simplifies things in your computer programming language, assume it's an integer.
One of the most common uses of Isqrt is in the division of an integer by all factors (or
primes) up to the
√ X of that
integer, either to find the factors of that integer, or to determine primality.
An alternative method for finding the Isqrt of a number is to
calculate: floor( sqrt(X) )
- where sqrt is the square root function for non─negative real numbers, and
- where floor is the floor function for real numbers.
If the hardware supports the computation of (real) square roots, the above method might be a faster method for
small numbers that don't have very many significant (decimal) digits.
However, floating point arithmetic is limited in the number of (binary or decimal) digits that it can support.
- Pseudo─code using quadratic residue
For this task, the integer square root of a non─negative number will be computed using a version of quadratic residue, which has the advantage that no floating point calculations are used, only integer arithmetic.
Furthermore, the two divisions can be performed by bit shifting, and the one multiplication can also be be performed by bit shifting or additions.
The disadvantage is the limitation of the size of the largest integer that a particular computer programming language can support.
Pseudo─code of a procedure for finding the integer square root of X (all variables are integers):
q ◄── 1 /*initialize Q to unity. */
/*find a power of 4 that's greater than X.*/
perform while q <= x /*perform while Q <= X. */
q ◄── q * 4 /*multiply Q by four. */
end /*perform*/
/*Q is now greater than X.*/
z ◄── x /*set Z to the value of X.*/
r ◄── 0 /*initialize R to zero. */
perform while q > 1 /*perform while Q > unity. */
q ◄── q ÷ 4 /*integer divide by four. */
t ◄── z - r - q /*compute value of T. */
r ◄── r ÷ 2 /*integer divide by two. */
if t >= 0 then do
z ◄── t /*set Z to value of T. */
r ◄── r + q /*compute new value of R. */
end
end /*perform*/
/*R is now the Isqrt(X). */
/* Sidenote: Also, Z is now the remainder after square root (i.e. */
/* R^2 + Z = X, so if Z = 0 then X is a perfect square). */
Another version for the (above) 1st perform is:
perform until q > X /*perform until Q > X. */
q ◄── q * 4 /*multiply Q by four. */
end /*perform*/
Integer square roots of some values:
Isqrt( 0) is 0 Isqrt(60) is 7 Isqrt( 99) is 9 Isqrt( 1) is 1 Isqrt(61) is 7 Isqrt(100) is 10 Isqrt( 2) is 1 Isqrt(62) is 7 Isqrt(102) is 10 Isqrt( 3) is 1 Isqrt(63) is 7 Isqrt( 4) is 2 Isqrt(64) is 8 Isqet(120) is 10 Isqrt( 5) is 2 Isqrt(65) is 8 Isqrt(121) is 11 Isqrt( 6) is 2 Isqrt(66) is 8 Isqrt(122) is 11 Isqrt( 7) is 2 Isqrt(67) is 8 Isqrt( 8) is 2 Isqrt(68) is 8 Isqrt(143) is 11 Isqrt( 9) is 3 Isqrt(69) is 8 Isqrt(144) is 12 Isqrt(10) is 3 Isqrt(70) is 8 Isqrt(145) is 12
- Task
Compute and show all output here (on this page) for:
- the
Isqrtof the integers from 0 ───► 65 (inclusive), shown in a horizontal format. - the
Isqrtof the odd powers from 71 ───► 773 (inclusive), shown in a vertical format. - use commas in the displaying of larger numbers.
- the
You can show more numbers for the 2nd requirement if the displays fits on one screen on Rosetta Code.
If your computer programming language only supports smaller integers, show what you can.
- Related tasks
Contents
- 1 11l
- 2 Ada
- 3 ALGOL 68
- 4 AppleScript
- 5 APL
- 6 Arturo
- 7 C
- 8 C++
- 9 C#
- 10 CLU
- 11 Cowgol
- 12 D
- 13 Delphi
- 14 F#
- 15 Factor
- 16 Forth
- 17 Fortran
- 18 FreeBASIC
- 19 Go
- 20 Haskell
- 21 J
- 22 Java
- 23 jq
- 24 Julia
- 25 Kotlin
- 26 Lua
- 27 MAD
- 28 Mathematica/Wolfram Language
- 29 Nim
- 30 Ol
- 31 Pascal
- 32 Perl
- 33 Phix
- 34 Python
- 35 Quackery
- 36 Racket
- 37 Raku
- 38 REXX
- 39 Ruby
- 40 Rust
- 41 Seed7
- 42 Sidef
- 43 Swift
- 44 Tiny BASIC
- 45 UNIX Shell
- 46 Visual Basic .NET
- 47 Wren
11l[edit]
F commatize(number, step = 3, sep = ‘,’)
V s = reversed(String(number))
String r = s[0]
L(i) 1 .< s.len
I i % step == 0
r ‘’= sep
r ‘’= s[i]
R reversed(r)
F isqrt(BigInt x)
assert(x >= 0)
V q = BigInt(1)
L q <= x
q *= 4
V z = x
V r = BigInt(0)
L q > 1
q I/= 4
V t = z - r - q
r I/= 2
I t >= 0
z = t
r += q
R r
print(‘The integer square root of integers from 0 to 65 are:’)
L(i) 66
print(isqrt(BigInt(i)), end' ‘ ’)
print()
print(‘The integer square roots of powers of 7 from 7^1 up to 7^73 are:’)
print(‘power 7 ^ power integer square root’)
print(‘----- --------------------------------------------------------------------------------- -----------------------------------------’)
V pow7 = BigInt(7)
V bi49 = BigInt(49)
L(i) (1..73).step(2)
print(‘#2 #84 #41’.format(i, commatize(pow7), commatize(isqrt(pow7))))
pow7 *= bi49
- Output:
The integer square root of integers from 0 to 65 are: 0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 The integer square roots of powers of 7 from 7^1 up to 7^73 are: power 7 ^ power integer square root ----- --------------------------------------------------------------------------------- ----------------------------------------- 1 7 2 3 343 18 5 16,807 129 7 823,543 907 9 40,353,607 6,352 11 1,977,326,743 44,467 13 96,889,010,407 311,269 15 4,747,561,509,943 2,178,889 17 232,630,513,987,207 15,252,229 19 11,398,895,185,373,143 106,765,608 21 558,545,864,083,284,007 747,359,260 23 27,368,747,340,080,916,343 5,231,514,822 25 1,341,068,619,663,964,900,807 36,620,603,758 27 65,712,362,363,534,280,139,543 256,344,226,312 29 3,219,905,755,813,179,726,837,607 1,794,409,584,184 31 157,775,382,034,845,806,615,042,743 12,560,867,089,291 33 7,730,993,719,707,444,524,137,094,407 87,926,069,625,040 35 378,818,692,265,664,781,682,717,625,943 615,482,487,375,282 37 18,562,115,921,017,574,302,453,163,671,207 4,308,377,411,626,977 39 909,543,680,129,861,140,820,205,019,889,143 30,158,641,881,388,842 41 44,567,640,326,363,195,900,190,045,974,568,007 211,110,493,169,721,897 43 2,183,814,375,991,796,599,109,312,252,753,832,343 1,477,773,452,188,053,281 45 107,006,904,423,598,033,356,356,300,384,937,784,807 10,344,414,165,316,372,973 47 5,243,338,316,756,303,634,461,458,718,861,951,455,543 72,410,899,157,214,610,812 49 256,923,577,521,058,878,088,611,477,224,235,621,321,607 506,876,294,100,502,275,687 51 12,589,255,298,531,885,026,341,962,383,987,545,444,758,743 3,548,134,058,703,515,929,815 53 616,873,509,628,062,366,290,756,156,815,389,726,793,178,407 24,836,938,410,924,611,508,707 55 30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943 173,858,568,876,472,280,560,953 57 1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207 1,217,009,982,135,305,963,926,677 59 72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143 8,519,069,874,947,141,747,486,745 61 3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007 59,633,489,124,629,992,232,407,216 63 174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343 417,434,423,872,409,945,626,850,517 65 8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807 2,922,040,967,106,869,619,387,953,625 67 418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543 20,454,286,769,748,087,335,715,675,381 69 20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607 143,180,007,388,236,611,350,009,727,669 71 1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 1,002,260,051,717,656,279,450,068,093,686 73 49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 7,015,820,362,023,593,956,150,476,655,802
Ada[edit]
with Ada.Text_Io;
with Ada.Numerics.Big_Numbers.Big_Integers;
with Ada.Strings.Fixed;
procedure Integer_Square_Root is
use Ada.Numerics.Big_Numbers.Big_Integers;
use Ada.Text_Io;
function Isqrt (X : Big_Integer) return Big_Integer is
Q : Big_Integer := 1;
Z, T, R : Big_Integer;
begin
while Q <= X loop
Q := Q * 4;
end loop;
Z := X;
R := 0;
while Q > 1 loop
Q := Q / 4;
T := Z - R - Q;
R := R / 2;
if T >= 0 then
Z := T;
R := R + Q;
end if;
end loop;
return R;
end Isqrt;
function Commatize (N : Big_Integer; Width : Positive) return String is
S : constant String := To_String (N, Width);
Image : String (1 .. Width + Width / 3) := (others => ' ');
Pos : Natural := Image'Last;
begin
for I in S'Range loop
Image (Pos) := S (S'Last - I + S'First);
exit when Image (Pos) = ' ';
Pos := Pos - 1;
if I mod 3 = 0 and S (S'Last - I + S'First - 1) /= ' ' then
Image (Pos) := ''';
Pos := Pos - 1;
end if;
end loop;
return Image;
end Commatize;
type Mode_Kind is (Tens, Ones, Spacer, Result);
begin
Put_Line ("Integer square roots of integers 0 .. 65:");
for Mode in Mode_Kind loop
for N in 0 .. 65 loop
case Mode is
when Tens => Put ((if N / 10 = 0
then " "
else Natural'Image (N / 10)));
when Ones => Put (Natural'Image (N mod 10));
when Spacer => Put ("--");
when Result => Put (To_String (Isqrt (To_Big_Integer (N))));
end case;
end loop;
New_Line;
end loop;
New_Line;
declare
package Integer_Io is new Ada.Text_Io.Integer_Io (Natural);
use Ada.Strings.Fixed;
N : Integer := 1;
P, R : Big_Integer;
begin
Put_Line ("| N|" & 80 * " " & "7**N|" & 30 * " " & "isqrt (7**N)|");
Put_Line (133 * "=");
loop
P := 7**N;
R := Isqrt (P);
Put ("|"); Integer_Io.Put (N, Width => 3);
Put ("|"); Put (Commatize (P, Width => 63));
Put ("|"); Put (Commatize (R, Width => 32));
Put ("|"); New_Line;
exit when N >= 73;
N := N + 2;
end loop;
Put_Line (133 * "=");
end;
end Integer_Square_Root;
- Output:
Integer square roots of integers 0 .. 65:
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
------------------------------------------------------------------------------------------------------------------------------------
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8
| N| 7**N| isqrt (7**N)|
=====================================================================================================================================
| 1| 7| 2|
| 3| 343| 18|
| 5| 16'807| 129|
| 7| 823'543| 907|
| 9| 40'353'607| 6'352|
| 11| 1'977'326'743| 44'467|
| 13| 96'889'010'407| 311'269|
| 15| 4'747'561'509'943| 2'178'889|
| 17| 232'630'513'987'207| 15'252'229|
| 19| 11'398'895'185'373'143| 106'765'608|
| 21| 558'545'864'083'284'007| 747'359'260|
| 23| 27'368'747'340'080'916'343| 5'231'514'822|
| 25| 1'341'068'619'663'964'900'807| 36'620'603'758|
| 27| 65'712'362'363'534'280'139'543| 256'344'226'312|
| 29| 3'219'905'755'813'179'726'837'607| 1'794'409'584'184|
| 31| 157'775'382'034'845'806'615'042'743| 12'560'867'089'291|
| 33| 7'730'993'719'707'444'524'137'094'407| 87'926'069'625'040|
| 35| 378'818'692'265'664'781'682'717'625'943| 615'482'487'375'282|
| 37| 18'562'115'921'017'574'302'453'163'671'207| 4'308'377'411'626'977|
| 39| 909'543'680'129'861'140'820'205'019'889'143| 30'158'641'881'388'842|
| 41| 44'567'640'326'363'195'900'190'045'974'568'007| 211'110'493'169'721'897|
| 43| 2'183'814'375'991'796'599'109'312'252'753'832'343| 1'477'773'452'188'053'281|
| 45| 107'006'904'423'598'033'356'356'300'384'937'784'807| 10'344'414'165'316'372'973|
| 47| 5'243'338'316'756'303'634'461'458'718'861'951'455'543| 72'410'899'157'214'610'812|
| 49| 256'923'577'521'058'878'088'611'477'224'235'621'321'607| 506'876'294'100'502'275'687|
| 51| 12'589'255'298'531'885'026'341'962'383'987'545'444'758'743| 3'548'134'058'703'515'929'815|
| 53| 616'873'509'628'062'366'290'756'156'815'389'726'793'178'407| 24'836'938'410'924'611'508'707|
| 55| 30'226'801'971'775'055'948'247'051'683'954'096'612'865'741'943| 173'858'568'876'472'280'560'953|
| 57| 1'481'113'296'616'977'741'464'105'532'513'750'734'030'421'355'207| 1'217'009'982'135'305'963'926'677|
| 59| 72'574'551'534'231'909'331'741'171'093'173'785'967'490'646'405'143| 8'519'069'874'947'141'747'486'745|
| 61| 3'556'153'025'177'363'557'255'317'383'565'515'512'407'041'673'852'007| 59'633'489'124'629'992'232'407'216|
| 63| 174'251'498'233'690'814'305'510'551'794'710'260'107'945'042'018'748'343| 417'434'423'872'409'945'626'850'517|
| 65| 8'538'323'413'450'849'900'970'017'037'940'802'745'289'307'058'918'668'807| 2'922'040'967'106'869'619'387'953'625|
| 67| 418'377'847'259'091'645'147'530'834'859'099'334'519'176'045'887'014'771'543| 20'454'286'769'748'087'335'715'675'381|
| 69| 20'500'514'515'695'490'612'229'010'908'095'867'391'439'626'248'463'723'805'607| 143'180'007'388'236'611'350'009'727'669|
| 71| 1'004'525'211'269'079'039'999'221'534'496'697'502'180'541'686'174'722'466'474'743| 1'002'260'051'717'656'279'450'068'093'686|
| 73| 49'221'735'352'184'872'959'961'855'190'338'177'606'846'542'622'561'400'857'262'407| 7'015'820'362'023'593'956'150'476'655'802|
=====================================================================================================================================
ALGOL 68[edit]
Implements the task pseudo-code.
BEGIN # Integer square roots #
PR precision 200 PR
# returns the integer square root of x; x must be >= 0 #
PROC isqrt = ( LONG LONG INT x )LONG LONG INT:
IF x < 0 THEN print( ( "Negative number in isqrt", newline ) );stop
ELIF x < 2 THEN x
ELSE
# x is greater than 1 #
# find a power of 4 that's greater than x #
LONG LONG INT q := 1;
WHILE q <= x DO q *:= 4 OD;
# find the root #
LONG LONG INT z := x;
LONG LONG INT r := 0;
WHILE q > 1 DO
q OVERAB 4;
LONG LONG INT t = z - r - q;
r OVERAB 2;
IF t >= 0 THEN
z := t;
r +:= q
FI
OD;
r
FI; # isqrt #
# returns a string representation of n with commas #
PROC commatise = ( LONG LONG INT n )STRING:
BEGIN
STRING result := "";
STRING unformatted = whole( n, 0 );
INT ch count := 0;
FOR c FROM UPB unformatted BY -1 TO LWB unformatted DO
IF ch count <= 2 THEN ch count +:= 1
ELSE ch count := 1; "," +=: result
FI;
unformatted[ c ] +=: result
OD;
result
END; # commatise #
# left-pads a string to at least n characters #
PROC pad left = ( STRING s, INT n )STRING:
BEGIN
STRING result := s;
WHILE ( UPB result - LWB result ) + 1 < n DO " " +=: result OD;
result
END; # pad left #
# task test cases #
print( ( "Integer square roots of 0..65", newline ) );
FOR i FROM 0 TO 65 DO print( ( " ", whole( isqrt( i ), 0 ) ) ) OD;
print( ( newline ) );
# integer square roots of odd powers of 7 #
print( ( "Integer square roots of 7^n", newline ) );
print( ( " n|", pad left( "7^n", 82 ), "|", pad left( "isqrt(7^n)", 42 ), newline ) );
LONG LONG INT p7 := 7;
FOR p BY 2 TO 73 DO
print( ( whole( p, -2 )
, "|"
, pad left( commatise( p7 ), 82 )
, "|"
, pad left( commatise( isqrt( p7 ) ), 42 )
, newline
)
);
p7 *:= 49
OD
END
- Output:
Integer square roots of 0..65 0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 Integer square roots of 7^n n| 7^n| isqrt(7^n) 1| 7| 2 3| 343| 18 5| 16,807| 129 7| 823,543| 907 9| 40,353,607| 6,352 11| 1,977,326,743| 44,467 13| 96,889,010,407| 311,269 15| 4,747,561,509,943| 2,178,889 17| 232,630,513,987,207| 15,252,229 19| 11,398,895,185,373,143| 106,765,608 21| 558,545,864,083,284,007| 747,359,260 23| 27,368,747,340,080,916,343| 5,231,514,822 25| 1,341,068,619,663,964,900,807| 36,620,603,758 27| 65,712,362,363,534,280,139,543| 256,344,226,312 29| 3,219,905,755,813,179,726,837,607| 1,794,409,584,184 31| 157,775,382,034,845,806,615,042,743| 12,560,867,089,291 33| 7,730,993,719,707,444,524,137,094,407| 87,926,069,625,040 35| 378,818,692,265,664,781,682,717,625,943| 615,482,487,375,282 37| 18,562,115,921,017,574,302,453,163,671,207| 4,308,377,411,626,977 39| 909,543,680,129,861,140,820,205,019,889,143| 30,158,641,881,388,842 41| 44,567,640,326,363,195,900,190,045,974,568,007| 211,110,493,169,721,897 43| 2,183,814,375,991,796,599,109,312,252,753,832,343| 1,477,773,452,188,053,281 45| 107,006,904,423,598,033,356,356,300,384,937,784,807| 10,344,414,165,316,372,973 47| 5,243,338,316,756,303,634,461,458,718,861,951,455,543| 72,410,899,157,214,610,812 49| 256,923,577,521,058,878,088,611,477,224,235,621,321,607| 506,876,294,100,502,275,687 51| 12,589,255,298,531,885,026,341,962,383,987,545,444,758,743| 3,548,134,058,703,515,929,815 53| 616,873,509,628,062,366,290,756,156,815,389,726,793,178,407| 24,836,938,410,924,611,508,707 55| 30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943| 173,858,568,876,472,280,560,953 57| 1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207| 1,217,009,982,135,305,963,926,677 59| 72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143| 8,519,069,874,947,141,747,486,745 61| 3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007| 59,633,489,124,629,992,232,407,216 63| 174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343| 417,434,423,872,409,945,626,850,517 65| 8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807| 2,922,040,967,106,869,619,387,953,625 67| 418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543| 20,454,286,769,748,087,335,715,675,381 69| 20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607| 143,180,007,388,236,611,350,009,727,669 71| 1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743| 1,002,260,051,717,656,279,450,068,093,686 73|49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407| 7,015,820,362,023,593,956,150,476,655,802
AppleScript[edit]
The odd-powers-of-7 part of the task is limited by the precision of AppleScript reals.
on isqrt(x)
set q to 1
repeat until (q > x)
set q to q * 4
end repeat
set z to x
set r to 0
repeat while (q > 1)
set q to q div 4
set t to z - r - q
set r to r div 2
if (t > -1) then
set z to t
set r to r + q
end if
end repeat
return r
end isqrt
-- Task code
on intToText(n, separator)
set output to ""
repeat until (n < 1000)
set output to separator & (text 2 thru 4 of ((1000 + (n mod 1000) as integer) as text)) & output
set n to n div 1000
end repeat
return (n as integer as text) & output
end intToText
on doTask()
-- Get the integer and power results.
set {integerResults, powerResults} to {{}, {}}
repeat with x from 0 to 65
set end of integerResults to isqrt(x)
end repeat
repeat with p from 1 to 73 by 2
set x to 7 ^ p
if (x > 1.0E+15) then exit repeat -- Beyond the precision of AppleScript reals.
set end of powerResults to "7^" & p & tab & "(" & intToText(x, ",") & "):" & (tab & tab & intToText(isqrt(x), ","))
end repeat
-- Format and output.
set astid to AppleScript's text item delimiters
set AppleScript's text item delimiters to space
set output to {"Isqrts of integers from 0 to 65:", space & integerResults, ¬
"Isqrts of odd powers of 7 from 1 to " & (p - 2) & ":", powerResults}
set AppleScript's text item delimiters to linefeed
set output to output as text
set AppleScript's text item delimiters to astid
return output
end doTask
doTask()
- Output:
"Isqrts of integers from 0 to 65:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8
Isqrts of odd powers of 7 from 1 to 17:
7^1 (7): 2
7^3 (343): 18
7^5 (16,807): 129
7^7 (823,543): 907
7^9 (40,353,607): 6,352
7^11 (1,977,326,743): 44,467
7^13 (96,889,010,407): 311,269
7^15 (4,747,561,509,943): 2,178,889
7^17 (232,630,513,987,207): 15,252,229"
APL[edit]
Works in Dyalog APL
i←{x←⍵
q←(×∘4)⍣{⍺>x}⊢1
⊃{ r z q←⍵
q←⌊q÷4
t←(z-r)-q
r←⌊r÷2
z←z t[1+t≥0]
r←r+q×t≥0
r z q
}⍣{ r z q←⍺
q≤1
}⊢0 x q
}
- Output:
i¨⍳65
1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6
6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8
(⎕fr⎕pp)←1287 34
↑{⍵ (7*⍵) (i 7*⍵)}¨1,1+2×⍳10
1 7 2
3 343 18
5 16807 129
7 823543 907
9 40353607 6352
11 1977326743 44467
13 96889010407 311269
15 4747561509943 2178889
17 232630513987207 15252229
19 11398895185373143 106765608
21 558545864083284007 747359260
Arturo[edit]
commatize: function [x][
reverse join.with:"," map split.every: 3 split reverse to :string x => join
]
isqrt: function [x][
num: new x
q: new 1
r: new 0
while [q =< num]-> shl.safe 'q 2
while [q > 1][
shr 'q 2
t: (num-r)-q
shr 'r 1
if t >= 0 [
num: t
r: new r+q
]
]
return r
]
print map 0..65 => isqrt
loop range 1 .step: 2 72 'n ->
print [n "\t" commatize isqrt 7^n]
- Output:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 1 2 3 18 5 129 7 907 9 6,352 11 44,467 13 311,269 15 2,178,889 17 15,252,229 19 106,765,608 21 747,359,260 23 5,231,514,822 25 36,620,603,758 27 256,344,226,312 29 1,794,409,584,184 31 12,560,867,089,291 33 87,926,069,625,040 35 615,482,487,375,282 37 4,308,377,411,626,977 39 30,158,641,881,388,842 41 211,110,493,169,721,897 43 1,477,773,452,188,053,281 45 10,344,414,165,316,372,973 47 72,410,899,157,214,610,812 49 506,876,294,100,502,275,687 51 3,548,134,058,703,515,929,815 53 24,836,938,410,924,611,508,707 55 173,858,568,876,472,280,560,953 57 1,217,009,982,135,305,963,926,677 59 8,519,069,874,947,141,747,486,745 61 59,633,489,124,629,992,232,407,216 63 417,434,423,872,409,945,626,850,517 65 2,922,040,967,106,869,619,387,953,625 67 20,454,286,769,748,087,335,715,675,381 69 143,180,007,388,236,611,350,009,727,669 71 1,002,260,051,717,656,279,450,068,093,686
C[edit]
Up to 64-bit limits with no big int library.
#include <stdint.h>
#include <stdio.h>
int64_t isqrt(int64_t x) {
int64_t q = 1, r = 0;
while (q <= x) {
q <<= 2;
}
while (q > 1) {
int64_t t;
q >>= 2;
t = x - r - q;
r >>= 1;
if (t >= 0) {
x = t;
r += q;
}
}
return r;
}
int main() {
int64_t p;
int n;
printf("Integer square root for numbers 0 to 65:\n");
for (n = 0; n <= 65; n++) {
printf("%lld ", isqrt(n));
}
printf("\n\n");
printf("Integer square roots of odd powers of 7 from 1 to 21:\n");
printf(" n | 7 ^ n | isqrt(7 ^ n)\n");
p = 7;
for (n = 1; n <= 21; n += 2, p *= 49) {
printf("%2d | %18lld | %12lld\n", n, p, isqrt(p));
}
}
- Output:
Integer square root for numbers 0 to 65: 0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 Integer square roots of odd powers of 7 from 1 to 21: n | 7 ^ n | isqrt(7 ^ n) 1 | 7 | 2 3 | 343 | 18 5 | 16807 | 129 7 | 823543 | 907 9 | 40353607 | 6352 11 | 1977326743 | 44467 13 | 96889010407 | 311269 15 | 4747561509943 | 2178889 17 | 232630513987207 | 15252229 19 | 11398895185373143 | 106765608 21 | 558545864083284007 | 747359260
C++[edit]
#include <iomanip>
#include <iostream>
#include <sstream>
#include <boost/multiprecision/cpp_int.hpp>
using big_int = boost::multiprecision::cpp_int;
template <typename integer>
integer isqrt(integer x) {
integer q = 1;
while (q <= x)
q <<= 2;
integer r = 0;
while (q > 1) {
q >>= 2;
integer t = x - r - q;
r >>= 1;
if (t >= 0) {
x = t;
r += q;
}
}
return r;
}
std::string commatize(const big_int& n) {
std::ostringstream out;
out << n;
std::string str(out.str());
std::string result;
size_t digits = str.size();
result.reserve(4 * digits/3);
for (size_t i = 0; i < digits; ++i) {
if (i > 0 && i % 3 == digits % 3)
result += ',';
result += str[i];
}
return result;
}
int main() {
std::cout << "Integer square root for numbers 0 to 65:\n";
for (int n = 0; n <= 65; ++n)
std::cout << isqrt(n) << ' ';
std::cout << "\n\n";
std::cout << "Integer square roots of odd powers of 7 from 1 to 73:\n";
const int power_width = 83, isqrt_width = 42;
std::cout << " n |"
<< std::setw(power_width) << "7 ^ n" << " |"
<< std::setw(isqrt_width) << "isqrt(7 ^ n)"
<< '\n';
std::cout << std::string(6 + power_width + isqrt_width, '-') << '\n';
big_int p = 7;
for (int n = 1; n <= 73; n += 2, p *= 49) {
std::cout << std::setw(2) << n << " |"
<< std::setw(power_width) << commatize(p) << " |"
<< std::setw(isqrt_width) << commatize(isqrt(p))
<< '\n';
}
return 0;
}
- Output:
Integer square root for numbers 0 to 65: 0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 Integer square roots of odd powers of 7 from 1 to 73: n | 7 ^ n | isqrt(7 ^ n) ----------------------------------------------------------------------------------------------------------------------------------- 1 | 7 | 2 3 | 343 | 18 5 | 16,807 | 129 7 | 823,543 | 907 9 | 40,353,607 | 6,352 11 | 1,977,326,743 | 44,467 13 | 96,889,010,407 | 311,269 15 | 4,747,561,509,943 | 2,178,889 17 | 232,630,513,987,207 | 15,252,229 19 | 11,398,895,185,373,143 | 106,765,608 21 | 558,545,864,083,284,007 | 747,359,260 23 | 27,368,747,340,080,916,343 | 5,231,514,822 25 | 1,341,068,619,663,964,900,807 | 36,620,603,758 27 | 65,712,362,363,534,280,139,543 | 256,344,226,312 29 | 3,219,905,755,813,179,726,837,607 | 1,794,409,584,184 31 | 157,775,382,034,845,806,615,042,743 | 12,560,867,089,291 33 | 7,730,993,719,707,444,524,137,094,407 | 87,926,069,625,040 35 | 378,818,692,265,664,781,682,717,625,943 | 615,482,487,375,282 37 | 18,562,115,921,017,574,302,453,163,671,207 | 4,308,377,411,626,977 39 | 909,543,680,129,861,140,820,205,019,889,143 | 30,158,641,881,388,842 41 | 44,567,640,326,363,195,900,190,045,974,568,007 | 211,110,493,169,721,897 43 | 2,183,814,375,991,796,599,109,312,252,753,832,343 | 1,477,773,452,188,053,281 45 | 107,006,904,423,598,033,356,356,300,384,937,784,807 | 10,344,414,165,316,372,973 47 | 5,243,338,316,756,303,634,461,458,718,861,951,455,543 | 72,410,899,157,214,610,812 49 | 256,923,577,521,058,878,088,611,477,224,235,621,321,607 | 506,876,294,100,502,275,687 51 | 12,589,255,298,531,885,026,341,962,383,987,545,444,758,743 | 3,548,134,058,703,515,929,815 53 | 616,873,509,628,062,366,290,756,156,815,389,726,793,178,407 | 24,836,938,410,924,611,508,707 55 | 30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943 | 173,858,568,876,472,280,560,953 57 | 1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207 | 1,217,009,982,135,305,963,926,677 59 | 72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143 | 8,519,069,874,947,141,747,486,745 61 | 3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007 | 59,633,489,124,629,992,232,407,216 63 | 174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343 | 417,434,423,872,409,945,626,850,517 65 | 8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807 | 2,922,040,967,106,869,619,387,953,625 67 | 418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543 | 20,454,286,769,748,087,335,715,675,381 69 | 20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607 | 143,180,007,388,236,611,350,009,727,669 71 | 1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 | 1,002,260,051,717,656,279,450,068,093,686 73 | 49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 | 7,015,820,362,023,593,956,150,476,655,802
C#[edit]
using System;
using static System.Console;
using BI = System.Numerics.BigInteger;
class Program {
static BI isqrt(BI x) { BI q = 1, r = 0, t; while (q <= x) q <<= 2; while (q > 1) {
q >>= 2; t = x - r - q; r >>= 1; if (t >= 0) { x = t; r += q; } } return r; }
static void Main() { const int max = 73, smax = 65;
int power_width = ((BI.Pow(7, max).ToString().Length / 3) << 2) + 3,
isqrt_width = (power_width + 1) >> 1;
WriteLine("Integer square root for numbers 0 to {0}:", smax);
for (int n = 0; n <= smax; ++n) Write("{0} ",
(n / 10).ToString().Replace("0", " ")); WriteLine();
for (int n = 0; n <= smax; ++n) Write("{0} ", n % 10); WriteLine();
WriteLine(new String('-', (smax << 1) + 1));
for (int n = 0; n <= smax; ++n) Write("{0} ", isqrt(n));
WriteLine("\n\nInteger square roots of odd powers of 7 from 1 to {0}:", max);
string s = string.Format("[0,2] |[1,{0}:n0] |[2,{1}:n0]",
power_width, isqrt_width).Replace("[", "{").Replace("]", "}");
WriteLine(s, "n", "7 ^ n", "isqrt(7 ^ n)");
WriteLine(new String('-', power_width + isqrt_width + 6));
BI p = 7; for (int n = 1; n <= max; n += 2, p *= 49)
WriteLine (s, n, p, isqrt(p)); }
}
- Output:
Integer square root for numbers 0 to 65:
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
-----------------------------------------------------------------------------------------------------------------------------------
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8
Integer square roots of odd powers of 7 from 1 to 73:
n | 7 ^ n | isqrt(7 ^ n)
-----------------------------------------------------------------------------------------------------------------------------------
1 | 7 | 2
3 | 343 | 18
5 | 16,807 | 129
7 | 823,543 | 907
9 | 40,353,607 | 6,352
11 | 1,977,326,743 | 44,467
13 | 96,889,010,407 | 311,269
15 | 4,747,561,509,943 | 2,178,889
17 | 232,630,513,987,207 | 15,252,229
19 | 11,398,895,185,373,143 | 106,765,608
21 | 558,545,864,083,284,007 | 747,359,260
23 | 27,368,747,340,080,916,343 | 5,231,514,822
25 | 1,341,068,619,663,964,900,807 | 36,620,603,758
27 | 65,712,362,363,534,280,139,543 | 256,344,226,312
29 | 3,219,905,755,813,179,726,837,607 | 1,794,409,584,184
31 | 157,775,382,034,845,806,615,042,743 | 12,560,867,089,291
33 | 7,730,993,719,707,444,524,137,094,407 | 87,926,069,625,040
35 | 378,818,692,265,664,781,682,717,625,943 | 615,482,487,375,282
37 | 18,562,115,921,017,574,302,453,163,671,207 | 4,308,377,411,626,977
39 | 909,543,680,129,861,140,820,205,019,889,143 | 30,158,641,881,388,842
41 | 44,567,640,326,363,195,900,190,045,974,568,007 | 211,110,493,169,721,897
43 | 2,183,814,375,991,796,599,109,312,252,753,832,343 | 1,477,773,452,188,053,281
45 | 107,006,904,423,598,033,356,356,300,384,937,784,807 | 10,344,414,165,316,372,973
47 | 5,243,338,316,756,303,634,461,458,718,861,951,455,543 | 72,410,899,157,214,610,812
49 | 256,923,577,521,058,878,088,611,477,224,235,621,321,607 | 506,876,294,100,502,275,687
51 | 12,589,255,298,531,885,026,341,962,383,987,545,444,758,743 | 3,548,134,058,703,515,929,815
53 | 616,873,509,628,062,366,290,756,156,815,389,726,793,178,407 | 24,836,938,410,924,611,508,707
55 | 30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943 | 173,858,568,876,472,280,560,953
57 | 1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207 | 1,217,009,982,135,305,963,926,677
59 | 72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143 | 8,519,069,874,947,141,747,486,745
61 | 3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007 | 59,633,489,124,629,992,232,407,216
63 | 174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343 | 417,434,423,872,409,945,626,850,517
65 | 8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807 | 2,922,040,967,106,869,619,387,953,625
67 | 418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543 | 20,454,286,769,748,087,335,715,675,381
69 | 20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607 | 143,180,007,388,236,611,350,009,727,669
71 | 1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 | 1,002,260,051,717,656,279,450,068,093,686
73 | 49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 | 7,015,820,362,023,593,956,150,476,655,802
CLU[edit]
% This program uses the 'bigint' cluster from PCLU's 'misc.lib'
% Integer square root of a bigint
isqrt = proc (x: bigint) returns (bigint)
% Initialize a couple of bigints we will reuse
own zero: bigint := bigint$i2bi(0)
own one: bigint := bigint$i2bi(1)
own two: bigint := bigint$i2bi(2)
own four: bigint := bigint$i2bi(4)
q: bigint := one
while q <= x do q := q * four end
t: bigint
z: bigint := x
r: bigint := zero
while q>one do
q := q / four
t := z - r - q
r := r / two
if t >= zero then
z := t
r := r + q
end
end
return(r)
end isqrt
% Format a bigint using commas
fmt = proc (x: bigint) returns (string)
own zero: bigint := bigint$i2bi(0)
own ten: bigint := bigint$i2bi(10)
if x=zero then return("0") end
out: array[char] := array[char]$[]
ds: int := 0
while x>zero do
array[char]$addl(out, char$i2c(bigint$bi2i(x // ten) + 48))
x := x / ten
ds := ds + 1
if x~=zero cand ds//3=0 then
array[char]$addl(out, ',')
end
end
return(string$ac2s(out))
end fmt
start_up = proc ()
po: stream := stream$primary_output()
% print square roots from 0..65
stream$putl(po, "isqrt of 0..65:")
for i: int in int$from_to(0, 65) do
stream$puts(po, fmt(isqrt(bigint$i2bi(i))) || " ")
end
% print square roots of odd powers
stream$putl(po, "\n\nisqrt of odd powers of 7:")
seven: bigint := bigint$i2bi(7)
for p: int in int$from_to_by(1, 73, 2) do
stream$puts(po, "isqrt(7^")
stream$putright(po, int$unparse(p), 2)
stream$puts(po, ") = ")
stream$putright(po, fmt(isqrt(seven ** bigint$i2bi(p))), 41)
stream$putl(po, "")
end
end start_up
- Output:
isqrt of 0..65: 0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 isqrt of odd powers of 7: isqrt(7^ 1) = 2 isqrt(7^ 3) = 18 isqrt(7^ 5) = 129 isqrt(7^ 7) = 907 isqrt(7^ 9) = 6,352 isqrt(7^11) = 44,467 isqrt(7^13) = 311,269 isqrt(7^15) = 2,178,889 isqrt(7^17) = 15,252,229 isqrt(7^19) = 106,765,608 isqrt(7^21) = 747,359,260 isqrt(7^23) = 5,231,514,822 isqrt(7^25) = 36,620,603,758 isqrt(7^27) = 256,344,226,312 isqrt(7^29) = 1,794,409,584,184 isqrt(7^31) = 12,560,867,089,291 isqrt(7^33) = 87,926,069,625,040 isqrt(7^35) = 615,482,487,375,282 isqrt(7^37) = 4,308,377,411,626,977 isqrt(7^39) = 30,158,641,881,388,842 isqrt(7^41) = 211,110,493,169,721,897 isqrt(7^43) = 1,477,773,452,188,053,281 isqrt(7^45) = 10,344,414,165,316,372,973 isqrt(7^47) = 72,410,899,157,214,610,812 isqrt(7^49) = 506,876,294,100,502,275,687 isqrt(7^51) = 3,548,134,058,703,515,929,815 isqrt(7^53) = 24,836,938,410,924,611,508,707 isqrt(7^55) = 173,858,568,876,472,280,560,953 isqrt(7^57) = 1,217,009,982,135,305,963,926,677 isqrt(7^59) = 8,519,069,874,947,141,747,486,745 isqrt(7^61) = 59,633,489,124,629,992,232,407,216 isqrt(7^63) = 417,434,423,872,409,945,626,850,517 isqrt(7^65) = 2,922,040,967,106,869,619,387,953,625 isqrt(7^67) = 20,454,286,769,748,087,335,715,675,381 isqrt(7^69) = 143,180,007,388,236,611,350,009,727,669 isqrt(7^71) = 1,002,260,051,717,656,279,450,068,093,686 isqrt(7^73) = 7,015,820,362,023,593,956,150,476,655,802
Cowgol[edit]
include "cowgol.coh";
# Integer square root
sub isqrt(x: uint32): (x0: uint32) is
x0 := x >> 1;
if x0 == 0 then
x0 := x;
return;
end if;
loop
var x1 := (x0 + x/x0) >> 1;
if x1 >= x0 then
break;
end if;
x0 := x1;
end loop;
end sub;
# Power
sub pow(x: uint32, n: uint8): (r: uint32) is
r := 1;
while n > 0 loop
r := r * x;
n := n - 1;
end loop;
end sub;
# Print integer square roots of 0..65
var n: uint32 := 0;
var col: uint8 := 11;
while n <= 65 loop
print_i32(isqrt(n));
col := col - 1;
if col == 0 then
print_nl();
col := 11;
else
print_char(' ');
end if;
n := n + 1;
end loop;
# Cowgol only supports 32-bit integers out of the box, so only powers of 7
# up to 7^11 are printed
var x: uint8 := 0;
while x <= 11 loop
print("isqrt(7^");
print_i8(x);
print(") = ");
print_i32(isqrt(pow(7, x)));
print_nl();
x := x + 1;
end loop;
- Output:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 isqrt(7^0) = 1 isqrt(7^1) = 2 isqrt(7^2) = 7 isqrt(7^3) = 18 isqrt(7^4) = 49 isqrt(7^5) = 129 isqrt(7^6) = 343 isqrt(7^7) = 907 isqrt(7^8) = 2401 isqrt(7^9) = 6352 isqrt(7^10) = 16807 isqrt(7^11) = 44467
D[edit]
import std.bigint;
import std.conv;
import std.exception;
import std.range;
import std.regex;
import std.stdio;
//Taken from the task http://rosettacode.org/wiki/Commatizing_numbers#D
auto commatize(in char[] txt, in uint start=0, in uint step=3, in string ins=",") @safe
in {
assert(step > 0);
} body {
if (start > txt.length || step > txt.length) {
return txt;
}
// First number may begin with digit or decimal point. Exponents ignored.
enum decFloField = ctRegex!("[0-9]*\\.[0-9]+|[0-9]+");
auto matchDec = matchFirst(txt[start .. $], decFloField);
if (!matchDec) {
return txt;
}
// Within a decimal float field:
// A decimal integer field to commatize is positive and not after a point.
enum decIntField = ctRegex!("(?<=\\.)|[1-9][0-9]*");
// A decimal fractional field is preceded by a point, and is only digits.
enum decFracField = ctRegex!("(?<=\\.)[0-9]+");
return txt[0 .. start] ~ matchDec.pre ~ matchDec.hit
.replace!(m => m.hit.retro.chunks(step).join(ins).retro)(decIntField)
.replace!(m => m.hit.chunks(step).join(ins))(decFracField)
~ matchDec.post;
}
auto commatize(BigInt v) {
return commatize(v.to!string);
}
BigInt sqrt(BigInt x) {
enforce(x >= 0);
auto q = BigInt(1);
while (q <= x) {
q <<= 2;
}
auto z = x;
auto r = BigInt(0);
while (q > 1) {
q >>= 2;
auto t = z;
t -= r;
t -= q;
r >>= 1;
if (t >= 0) {
z = t;
r += q;
}
}
return r;
}
void main() {
writeln("The integer square root of integers from 0 to 65 are:");
foreach (i; 0..66) {
write(sqrt(BigInt(i)), ' ');
}
writeln;
writeln("The integer square roots of powers of 7 from 7^1 up to 7^73 are:");
writeln("power 7 ^ power integer square root");
writeln("----- --------------------------------------------------------------------------------- -----------------------------------------");
auto pow7 = BigInt(7);
immutable bi49 = BigInt(49);
for (int i = 1; i <= 73; i += 2) {
writefln("%2d %84s %41s", i, pow7.commatize, sqrt(pow7).commatize);
pow7 *= bi49;
}
}
- Output:
The integer square root of integers from 0 to 65 are: 0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 The integer square roots of powers of 7 from 7^1 up to 7^73 are: power 7 ^ power integer square root ----- --------------------------------------------------------------------------------- ----------------------------------------- 1 7 2 3 343 18 5 16,807 129 7 823,543 907 9 40,353,607 6,352 11 1,977,326,743 44,467 13 96,889,010,407 311,269 15 4,747,561,509,943 2,178,889 17 232,630,513,987,207 15,252,229 19 11,398,895,185,373,143 106,765,608 21 558,545,864,083,284,007 747,359,260 23 27,368,747,340,080,916,343 5,231,514,822 25 1,341,068,619,663,964,900,807 36,620,603,758 27 65,712,362,363,534,280,139,543 256,344,226,312 29 3,219,905,755,813,179,726,837,607 1,794,409,584,184 31 157,775,382,034,845,806,615,042,743 12,560,867,089,291 33 7,730,993,719,707,444,524,137,094,407 87,926,069,625,040 35 378,818,692,265,664,781,682,717,625,943 615,482,487,375,282 37 18,562,115,921,017,574,302,453,163,671,207 4,308,377,411,626,977 39 909,543,680,129,861,140,820,205,019,889,143 30,158,641,881,388,842 41 44,567,640,326,363,195,900,190,045,974,568,007 211,110,493,169,721,897 43 2,183,814,375,991,796,599,109,312,252,753,832,343 1,477,773,452,188,053,281 45 107,006,904,423,598,033,356,356,300,384,937,784,807 10,344,414,165,316,372,973 47 5,243,338,316,756,303,634,461,458,718,861,951,455,543 72,410,899,157,214,610,812 49 256,923,577,521,058,878,088,611,477,224,235,621,321,607 506,876,294,100,502,275,687 51 12,589,255,298,531,885,026,341,962,383,987,545,444,758,743 3,548,134,058,703,515,929,815 53 616,873,509,628,062,366,290,756,156,815,389,726,793,178,407 24,836,938,410,924,611,508,707 55 30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943 173,858,568,876,472,280,560,953 57 1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207 1,217,009,982,135,305,963,926,677 59 72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143 8,519,069,874,947,141,747,486,745 61 3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007 59,633,489,124,629,992,232,407,216 63 174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343 417,434,423,872,409,945,626,850,517 65 8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807 2,922,040,967,106,869,619,387,953,625 67 418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543 20,454,286,769,748,087,335,715,675,381 69 20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607 143,180,007,388,236,611,350,009,727,669 71 1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 1,002,260,051,717,656,279,450,068,093,686 73 49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 7,015,820,362,023,593,956,150,476,655,802
Delphi[edit]
See #Pascal.
F#[edit]
// Find Integer Floor sqrt of a Large Integer. Nigel Galloway: July 17th., 2020
let Isqrt n=let rec fN i g l=match(l>0I,i-g-l) with
(true,e) when e>=0I->fN e (g/2I+l) (l/4I)
|(true,_) ->fN i (g/2I) (l/4I)
|_ ->g
fN n 0I (let rec fG g=if g>n then g/4I else fG (g*4I) in fG 1I)
[0I..65I]|>Seq.iter(Isqrt>>string>>printf "%s "); printfn "\n"
let fN n=7I**n in [1..2..73]|>Seq.iter(fN>>Isqrt>>printfn "%a" (fun n g -> n.Write("{0:#,#}", g)))
- Output:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 2 18 129 907 6,352 44,467 311,269 2,178,889 15,252,229 106,765,608 747,359,260 5,231,514,822 36,620,603,758 256,344,226,312 1,794,409,584,184 12,560,867,089,291 87,926,069,625,040 615,482,487,375,282 4,308,377,411,626,977 30,158,641,881,388,842 211,110,493,169,721,897 1,477,773,452,188,053,281 10,344,414,165,316,372,973 72,410,899,157,214,610,812 506,876,294,100,502,275,687 3,548,134,058,703,515,929,815 24,836,938,410,924,611,508,707 173,858,568,876,472,280,560,953 1,217,009,982,135,305,963,926,677 8,519,069,874,947,141,747,486,745 59,633,489,124,629,992,232,407,216 417,434,423,872,409,945,626,850,517 2,922,040,967,106,869,619,387,953,625 20,454,286,769,748,087,335,715,675,381 143,180,007,388,236,611,350,009,727,669 1,002,260,051,717,656,279,450,068,093,686 7,015,820,362,023,593,956,150,476,655,802
Factor[edit]
The isqrt word is a straightforward translation of the pseudocode from the task description using lexical variables.
USING: formatting io kernel locals math math.functions
math.ranges prettyprint sequences tools.memory.private ;
:: isqrt ( x -- n )
1 :> q!
[ q x > ] [ q 4 * q! ] until
x 0 :> ( z! r! )
[ q 1 > ] [
q 4 /i q!
z r - q - :> t
r -1 shift r!
t 0 >= [
t z!
r q + r!
] when
] while
r ;
"Integer square root for numbers 0 to 65 (inclusive):" print
66 <iota> [ bl ] [ isqrt pprint ] interleave nl nl
: align ( str str str -- ) "%2s%85s%44s\n" printf ;
: show ( n -- ) dup 7 swap ^ dup isqrt [ commas ] [email protected] align ;
"x" "7^x" "isqrt(7^x)" align
"-" "---" "----------" align
1 73 2 <range> [ show ] each
- Output:
Integer square root for numbers 0 to 65 (inclusive): 0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 x 7^x isqrt(7^x) - --- ---------- 1 7 2 3 343 18 5 16,807 129 7 823,543 907 9 40,353,607 6,352 11 1,977,326,743 44,467 13 96,889,010,407 311,269 15 4,747,561,509,943 2,178,889 17 232,630,513,987,207 15,252,229 19 11,398,895,185,373,143 106,765,608 21 558,545,864,083,284,007 747,359,260 23 27,368,747,340,080,916,343 5,231,514,822 25 1,341,068,619,663,964,900,807 36,620,603,758 27 65,712,362,363,534,280,139,543 256,344,226,312 29 3,219,905,755,813,179,726,837,607 1,794,409,584,184 31 157,775,382,034,845,806,615,042,743 12,560,867,089,291 33 7,730,993,719,707,444,524,137,094,407 87,926,069,625,040 35 378,818,692,265,664,781,682,717,625,943 615,482,487,375,282 37 18,562,115,921,017,574,302,453,163,671,207 4,308,377,411,626,977 39 909,543,680,129,861,140,820,205,019,889,143 30,158,641,881,388,842 41 44,567,640,326,363,195,900,190,045,974,568,007 211,110,493,169,721,897 43 2,183,814,375,991,796,599,109,312,252,753,832,343 1,477,773,452,188,053,281 45 107,006,904,423,598,033,356,356,300,384,937,784,807 10,344,414,165,316,372,973 47 5,243,338,316,756,303,634,461,458,718,861,951,455,543 72,410,899,157,214,610,812 49 256,923,577,521,058,878,088,611,477,224,235,621,321,607 506,876,294,100,502,275,687 51 12,589,255,298,531,885,026,341,962,383,987,545,444,758,743 3,548,134,058,703,515,929,815 53 616,873,509,628,062,366,290,756,156,815,389,726,793,178,407 24,836,938,410,924,611,508,707 55 30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943 173,858,568,876,472,280,560,953 57 1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207 1,217,009,982,135,305,963,926,677 59 72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143 8,519,069,874,947,141,747,486,745 61 3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007 59,633,489,124,629,992,232,407,216 63 174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343 417,434,423,872,409,945,626,850,517 65 8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807 2,922,040,967,106,869,619,387,953,625 67 418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543 20,454,286,769,748,087,335,715,675,381 69 20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607 143,180,007,388,236,611,350,009,727,669 71 1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 1,002,260,051,717,656,279,450,068,093,686 73 49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 7,015,820,362,023,593,956,150,476,655,802
Forth[edit]
Only handles odd powers of 7 up to 7^21.
: d., ( n -- ) \ write double precision int, commatized.
tuck dabs
<# begin 2dup 1.000 d> while # # # [char] , hold repeat #s rot sign #>
type space ;
: ., ( n -- ) \ write integer commatized.
s>d d., ;
: 4* s" 2 lshift" evaluate ; immediate
: 4/ s" 2 rshift" evaluate ; immediate
: isqrt-mod ( n -- z r ) \ n = r^2 + z
1 begin 2dup >= while 4* repeat
0 locals| r q z |
begin q 1 > while
q 4/ to q
z r - q - \ t
r 2/ to r
dup 0>= if
to z
r q + to r
else
drop
then
repeat z r ;
: isqrt isqrt-mod nip ;
: task1
." Integer square roots from 0 to 65:" cr
66 0 do i isqrt . loop cr ;
: task2
." Integer square roots of 7^n" cr
7 11 0 do
i 2* 1+ 2 .r 3 spaces
dup isqrt ., cr
49 *
loop ;
task1 cr task2 bye
This version of the core word does not require locals.
: sqrt-rem ( n -- sqrt rem)
>r 0 1 begin dup [email protected] > 0= while 4 * repeat
begin \ find a power of 4 greater than TORS
dup 1 > \ compute while greater than unity
while
2/ 2/ swap over over + negate [email protected] + \ integer divide by 4
dup 0< if drop 2/ else r> drop >r 2/ over + then swap
repeat drop r> ( sqrt rem)
;
: isqrt-mod sqrt-rem swap ;
- Output:
Integer square roots from 0 to 65: 0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 Integer square roots of 7^n 1 2 3 18 5 129 7 907 9 6,352 11 44,467 13 311,269 15 2,178,889 17 15,252,229 19 106,765,608 21 747,359,260
Fortran[edit]
MODULE INTEGER_SQUARE_ROOT
IMPLICIT NONE
CONTAINS
! Convert string representation number to string with comma digit separation
FUNCTION COMMATIZE(NUM) RESULT(OUT_STR)
INTEGER(16), INTENT(IN) :: NUM
INTEGER(16) I
CHARACTER(83) :: TEMP, OUT_STR
WRITE(TEMP, '(I0)') NUM
OUT_STR = ""
DO I=0, LEN_TRIM(TEMP)-1
IF (MOD(I, 3) .EQ. 0 .AND. I .GT. 0 .AND. I .LT. LEN_TRIM(TEMP)) THEN
OUT_STR = "," // TRIM(OUT_STR)
END IF
OUT_STR = TEMP(LEN_TRIM(TEMP)-I:LEN_TRIM(TEMP)-I) // TRIM(OUT_STR)
END DO
END FUNCTION COMMATIZE
! Calculate the integer square root for a given integer
FUNCTION ISQRT(NUM)
INTEGER(16), INTENT(IN) :: NUM
INTEGER(16) :: ISQRT
INTEGER(16) :: Q, Z, R, T
Q = 1
Z = NUM
R = 0
T = 0
DO WHILE (Q .LT. NUM)
Q = Q * 4
END DO
DO WHILE (Q .GT. 1)
Q = Q / 4
T = Z - R - Q
R = R / 2
IF (T .GE. 0) THEN
Z = T
R = R + Q
END IF
END DO
ISQRT = R
END FUNCTION ISQRT
END MODULE INTEGER_SQUARE_ROOT
! Demonstration of integer square root for numbers 0-65 followed by odd powers of 7
! from 1-73. Currently this demo takes significant time for numbers above 43
PROGRAM ISQRT_DEMO
USE INTEGER_SQUARE_ROOT
IMPLICIT NONE
INTEGER(16), PARAMETER :: MIN_NUM_HZ = 0
INTEGER(16), PARAMETER :: MAX_NUM_HZ = 65
INTEGER(16), PARAMETER :: POWER_BASE = 7
INTEGER(16), PARAMETER :: POWER_MIN = 1
INTEGER(16), PARAMETER :: POWER_MAX = 73
INTEGER(16), DIMENSION(MAX_NUM_HZ-MIN_NUM_HZ+1) :: VALUES
CHARACTER(2) :: HEADER_1
CHARACTER(83) :: HEADER_2
CHARACTER(83) :: HEADER_3
INTEGER(16) :: I
HEADER_1 = " n"
HEADER_2 = "7^n"
HEADER_3 = "isqrt(7^n)"
WRITE(*,'(A, I0, A, I0)') "Integer square root for numbers ", MIN_NUM_HZ, " to ", MAX_NUM_HZ
DO I=1, SIZE(VALUES)
VALUES(I) = ISQRT(MIN_NUM_HZ+I)
END DO
WRITE(*,'(100I2)') VALUES
WRITE(*,*) NEW_LINE('A')
WRITE(*,'(A,A,A,A,A)') HEADER_1, " | ", HEADER_2, " | ", HEADER_3
WRITE(*,*) REPEAT("-", 8+83*2)
DO I=POWER_MIN,POWER_MAX, 2
WRITE(*,'(I2, A, A, A, A)') I, " | " // COMMATIZE(7**I), " | ", COMMATIZE(ISQRT(7**I))
END DO
END PROGRAM ISQRT_DEMO
Integer square root for numbers 0 to 65 0 1 1 1 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 n | 7^n | isqrt(7^n) ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ 1 | 7 | 2 3 | 343 | 18 5 | 16,807 | 129 7 | 823,543 | 907 9 | 40,353,607 | 6,352 11 | 1,977,326,743 | 44,467 13 | 96,889,010,407 | 311,269 15 | 4,747,561,509,943 | 2,178,889 17 | 232,630,513,987,207 | 15,252,229 19 | 11,398,895,185,373,143 | 106,765,608 21 | 558,545,864,083,284,007 | 747,359,260 23 | 27,368,747,340,080,916,343 | 5,231,514,822 25 | 1,341,068,619,663,964,900,807 | 36,620,603,758 27 | 65,712,362,363,534,280,139,543 | 256,344,226,312 29 | 3,219,905,755,813,179,726,837,607 | 1,794,409,584,184 31 | 157,775,382,034,845,806,615,042,743 | 12,560,867,089,291 33 | 7,730,993,719,707,444,524,137,094,407 | 87,926,069,625,040 35 | 378,818,692,265,664,781,682,717,625,943 | 615,482,487,375,282 37 | 18,562,115,921,017,574,302,453,163,671,207 | 4,308,377,411,626,977 39 | 909,543,680,129,861,140,820,205,019,889,143 | 30,158,641,881,388,842 41 | 44,567,640,326,363,195,900,190,045,974,568,007 | 211,110,493,169,721,897 43 | 2,183,814,375,991,796,599,109,312,252,753,832,343 | 1,477,773,452,188,053,281
FreeBASIC[edit]
Odd powers up to 7^21 are shown; more would require an arbitrary precision library that would just add bloat without being illustrative.
function isqrt( byval x as ulongint ) as ulongint
dim as ulongint q = 1, r
dim as longint t
while q <= x
q = q shl 2
wend
while q > 1
q = q shr 2
t = x - r - q
r = r shr 1
if t >= 0 then
x = t
r += q
end if
wend
return r
end function
function commatize( byval N as string ) as string
dim as string bloat = ""
dim as uinteger c = 0
while N<>""
bloat = right(N,1) + bloat
N = left(N, len(N)-1)
c += 1
if c mod 3 = 0 and N<>"" then bloat = "," + bloat
wend
return bloat
end function
for i as ulongint = 0 to 65
print isqrt(i);" ";
next i
dim as string ns
for i as uinteger = 1 to 22 step 2
ns = str(isqrt(7^i))
print i, commatize(ns)
next i
- Output:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 1 2 3 18 5 129 7 907 9 6,352 11 44,467 13 311,269 15 2,178,889 17 15,252,229 19 106,765,608 21 747,359,260
Go[edit]
Go's big.Int type already has a built-in integer square root function but, as the point of this task appears to be to compute it using a particular algorithm, we re-code it from the pseudo-code given in the task description.
package main
import (
"fmt"
"log"
"math/big"
)
var zero = big.NewInt(0)
var one = big.NewInt(1)
func isqrt(x *big.Int) *big.Int {
if x.Cmp(zero) < 0 {
log.Fatal("Argument cannot be negative.")
}
q := big.NewInt(1)
for q.Cmp(x) <= 0 {
q.Lsh(q, 2)
}
z := new(big.Int).Set(x)
r := big.NewInt(0)
for q.Cmp(one) > 0 {
q.Rsh(q, 2)
t := new(big.Int)
t.Add(t, z)
t.Sub(t, r)
t.Sub(t, q)
r.Rsh(r, 1)
if t.Cmp(zero) >= 0 {
z.Set(t)
r.Add(r, q)
}
}
return r
}
func commatize(s string) string {
le := len(s)
for i := le - 3; i >= 1; i -= 3 {
s = s[0:i] + "," + s[i:]
}
return s
}
func main() {
fmt.Println("The integer square roots of integers from 0 to 65 are:")
for i := int64(0); i <= 65; i++ {
fmt.Printf("%d ", isqrt(big.NewInt(i)))
}
fmt.Println()
fmt.Println("\nThe integer square roots of powers of 7 from 7^1 up to 7^73 are:\n")
fmt.Println("power 7 ^ power integer square root")
fmt.Println("----- --------------------------------------------------------------------------------- -----------------------------------------")
pow7 := big.NewInt(7)
bi49 := big.NewInt(49)
for i := 1; i <= 73; i += 2 {
fmt.Printf("%2d %84s %41s\n", i, commatize(pow7.String()), commatize(isqrt(pow7).String()))
pow7.Mul(pow7, bi49)
}
}
- Output:
The integer square roots of integers from 0 to 65 are: 0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 The integer square roots of powers of 7 from 7^1 up to 7^73 are: power 7 ^ power integer square root ----- --------------------------------------------------------------------------------- ----------------------------------------- 1 7 2 3 343 18 5 16,807 129 7 823,543 907 9 40,353,607 6,352 11 1,977,326,743 44,467 13 96,889,010,407 311,269 15 4,747,561,509,943 2,178,889 17 232,630,513,987,207 15,252,229 19 11,398,895,185,373,143 106,765,608 21 558,545,864,083,284,007 747,359,260 23 27,368,747,340,080,916,343 5,231,514,822 25 1,341,068,619,663,964,900,807 36,620,603,758 27 65,712,362,363,534,280,139,543 256,344,226,312 29 3,219,905,755,813,179,726,837,607 1,794,409,584,184 31 157,775,382,034,845,806,615,042,743 12,560,867,089,291 33 7,730,993,719,707,444,524,137,094,407 87,926,069,625,040 35 378,818,692,265,664,781,682,717,625,943 615,482,487,375,282 37 18,562,115,921,017,574,302,453,163,671,207 4,308,377,411,626,977 39 909,543,680,129,861,140,820,205,019,889,143 30,158,641,881,388,842 41 44,567,640,326,363,195,900,190,045,974,568,007 211,110,493,169,721,897 43 2,183,814,375,991,796,599,109,312,252,753,832,343 1,477,773,452,188,053,281 45 107,006,904,423,598,033,356,356,300,384,937,784,807 10,344,414,165,316,372,973 47 5,243,338,316,756,303,634,461,458,718,861,951,455,543 72,410,899,157,214,610,812 49 256,923,577,521,058,878,088,611,477,224,235,621,321,607 506,876,294,100,502,275,687 51 12,589,255,298,531,885,026,341,962,383,987,545,444,758,743 3,548,134,058,703,515,929,815 53 616,873,509,628,062,366,290,756,156,815,389,726,793,178,407 24,836,938,410,924,611,508,707 55 30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943 173,858,568,876,472,280,560,953 57 1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207 1,217,009,982,135,305,963,926,677 59 72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143 8,519,069,874,947,141,747,486,745 61 3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007 59,633,489,124,629,992,232,407,216 63 174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343 417,434,423,872,409,945,626,850,517 65 8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807 2,922,040,967,106,869,619,387,953,625 67 418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543 20,454,286,769,748,087,335,715,675,381 69 20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607 143,180,007,388,236,611,350,009,727,669 71 1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 1,002,260,051,717,656,279,450,068,093,686 73 49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 7,015,820,362,023,593,956,150,476,655,802
Haskell[edit]
import Data.Bits
isqrt :: Integer -> Integer
isqrt n = go n 0 (q `shiftR` 2)
where
q = head $ dropWhile (< n) $ iterate (`shiftL` 2) 1
go z r 0 = r
go z r q = let t = z - r - q
in if t >= 0
then go t (r `shiftR` 1 + q) (q `shiftR` 2)
else go z (r `shiftR` 1) (q `shiftR` 2)
main = do
print $ isqrt <$> [1..65]
mapM_ print $ zip [1,3..73] (isqrt <$> iterate (49 *) 7)
*Main> main [0,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8] (1,2) (3,18) (5,129) (7,907) (9,6352) (11,44467) (13,311269) (15,2178889) (17,15252229) (19,106765608) (21,747359260) (23,5231514822) (25,36620603758) (27,256344226312) (29,1794409584184) (31,12560867089291) (33,87926069625040) (35,615482487375282) (37,4308377411626977) (39,30158641881388842) (41,211110493169721897) (43,1477773452188053281) (45,10344414165316372973) (47,72410899157214610812) (49,506876294100502275687) (51,3548134058703515929815) (53,24836938410924611508707) (55,173858568876472280560953) (57,1217009982135305963926677) (59,8519069874947141747486745) (61,59633489124629992232407216) (63,417434423872409945626850517) (65,2922040967106869619387953625) (67,20454286769748087335715675381) (69,143180007388236611350009727669) (71,1002260051717656279450068093686) (73,7015820362023593956150476655802)
J[edit]
Three implementations given. The floating point method is best for small square roots, Newton's method is fastest for extended integers. isqrt adapted from the page preamble.
isqrt_float=: <[email protected]:%:
isqrt_newton=: 9&$: :(x:@:<[email protected]:-:@:(] + x:@:<[email protected]:%)^:_&>~&:x:)
align=: (|.~ i.&' ')"1
comma=: (' ' -.~ [: }: [: , [: (|.) _3 (',' ,~ |.)\ |.)@":&>
While=: {{ u^:(0[email protected]:-:v)^:_ }}
isqrt=: 3 :0&>
y =. x: y
NB. q is a power of 4 that's greater than y. Append 0 0 under binary representation
q =. y (,&0 0x&.:#:@:])While>: 1x
z =. y NB. set z to the value of y.
r =. 0x NB. initialize r to zero.
while. 1 < q do. NB. perform while q > unity.
q =. _2&}.&.:#: q NB. integer divide by 4 (-2 drop under binary representation)
t =. (z - r) - q NB. compute value of t.
r =. }:&.:#: r NB. integer divide by two. (curtail under binary representation)
if. 0 <: t do.
z =. t NB. set z to value of t
r =. r + q NB. compute new value of r
end.
end.
NB. r is now the isqrt(y). (most recent value computed)
NB. Sidenote: Also, Z is now the remainder after square root
NB. ie. r^2 + z = y, so if z = 0 then x is a perfect square
NB. r , z
)
(,. isqrt_float) 7x ^ 20 21x
79792266297612001 282475249
558545864083284007 747359260
(,. isqrt_newton) 7x ^ 20 21x
79792266297612001 282475249
558545864083284007 747359260
align comma (,. isqrt) 7 ^&x: 1 2 p. i. 37
7
2
343
18
16,807
129
823,543
907
40,353,607
6,352
1,977,326,743
44,467
96,889,010,407
311,269
4,747,561,509,943
2,178,889
232,630,513,987,207
15,252,229
11,398,895,185,373,143
106,765,608
558,545,864,083,284,007
747,359,260
27,368,747,340,080,916,343
5,231,514,822
1,341,068,619,663,964,900,807
36,620,603,758
65,712,362,363,534,280,139,543
256,344,226,312
3,219,905,755,813,179,726,837,607
1,794,409,584,184
157,775,382,034,845,806,615,042,743
12,560,867,089,291
7,730,993,719,707,444,524,137,094,407
87,926,069,625,040
378,818,692,265,664,781,682,717,625,943
615,482,487,375,282
18,562,115,921,017,574,302,453,163,671,207
4,308,377,411,626,977
909,543,680,129,861,140,820,205,019,889,143
30,158,641,881,388,842
44,567,640,326,363,195,900,190,045,974,568,007
211,110,493,169,721,897
2,183,814,375,991,796,599,109,312,252,753,832,343
1,477,773,452,188,053,281
107,006,904,423,598,033,356,356,300,384,937,784,807
10,344,414,165,316,372,973
5,243,338,316,756,303,634,461,458,718,861,951,455,543
72,410,899,157,214,610,812
256,923,577,521,058,878,088,611,477,224,235,621,321,607
506,876,294,100,502,275,687
12,589,255,298,531,885,026,341,962,383,987,545,444,758,743
3,548,134,058,703,515,929,815
616,873,509,628,062,366,290,756,156,815,389,726,793,178,407
24,836,938,410,924,611,508,707
30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943
173,858,568,876,472,280,560,953
1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207
1,217,009,982,135,305,963,926,677
72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143
8,519,069,874,947,141,747,486,745
3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007
59,633,489,124,629,992,232,407,216
174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343
417,434,423,872,409,945,626,850,517
8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807
2,922,040,967,106,869,619,387,953,625
418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543
20,454,286,769,748,087,335,715,675,381
20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607
143,180,007,388,236,611,350,009,727,669
1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743
1,002,260,051,717,656,279,450,068,093,686
49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407
7,015,820,362,023,593,956,150,476,655,802
NB. Newton's method result matches isqrt
(isqrt_newton -: isqrt)7 ^&x: 1 2 p. i. 37
1
NB. An order of magnitude faster and one tenth the space, in j
timespacex 'isqrt_newton 7 ^&x: 1 2 p. i. 37'
0.038085 39552
timespacex 'isqrt 7 ^&x: 1 2 p. i. 37'
0.367744 319712
Java[edit]
import java.math.BigInteger;
public class Isqrt {
private static BigInteger isqrt(BigInteger x) {
if (x.compareTo(BigInteger.ZERO) < 0) {
throw new IllegalArgumentException("Argument cannot be negative");
}
var q = BigInteger.ONE;
while (q.compareTo(x) <= 0) {
q = q.shiftLeft(2);
}
var z = x;
var r = BigInteger.ZERO;
while (q.compareTo(BigInteger.ONE) > 0) {
q = q.shiftRight(2);
var t = z;
t = t.subtract(r);
t = t.subtract(q);
r = r.shiftRight(1);
if (t.compareTo(BigInteger.ZERO) >= 0) {
z = t;
r = r.add(q);
}
}
return r;
}
public static void main(String[] args) {
System.out.println("The integer square root of integers from 0 to 65 are:");
for (int i = 0; i <= 65; i++) {
System.out.printf("%s ", isqrt(BigInteger.valueOf(i)));
}
System.out.println();
System.out.println("The integer square roots of powers of 7 from 7^1 up to 7^73 are:");
System.out.println("power 7 ^ power integer square root");
System.out.println("----- --------------------------------------------------------------------------------- -----------------------------------------");
var pow7 = BigInteger.valueOf(7);
var bi49 = BigInteger.valueOf(49);
for (int i = 1; i < 74; i += 2) {
System.out.printf("%2d %,84d %,41d\n", i, pow7, isqrt(pow7));
pow7 = pow7.multiply(bi49);
}
}
}
- Output:
The integer square root of integers from 0 to 65 are: 0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 The integer square roots of powers of 7 from 7^1 up to 7^73 are: power 7 ^ power integer square root ----- --------------------------------------------------------------------------------- ----------------------------------------- 1 7 2 3 343 18 5 16,807 129 7 823,543 907 9 40,353,607 6,352 11 1,977,326,743 44,467 13 96,889,010,407 311,269 15 4,747,561,509,943 2,178,889 17 232,630,513,987,207 15,252,229 19 11,398,895,185,373,143 106,765,608 21 558,545,864,083,284,007 747,359,260 23 27,368,747,340,080,916,343 5,231,514,822 25 1,341,068,619,663,964,900,807 36,620,603,758 27 65,712,362,363,534,280,139,543 256,344,226,312 29 3,219,905,755,813,179,726,837,607 1,794,409,584,184 31 157,775,382,034,845,806,615,042,743 12,560,867,089,291 33 7,730,993,719,707,444,524,137,094,407 87,926,069,625,040 35 378,818,692,265,664,781,682,717,625,943 615,482,487,375,282 37 18,562,115,921,017,574,302,453,163,671,207 4,308,377,411,626,977 39 909,543,680,129,861,140,820,205,019,889,143 30,158,641,881,388,842 41 44,567,640,326,363,195,900,190,045,974,568,007 211,110,493,169,721,897 43 2,183,814,375,991,796,599,109,312,252,753,832,343 1,477,773,452,188,053,281 45 107,006,904,423,598,033,356,356,300,384,937,784,807 10,344,414,165,316,372,973 47 5,243,338,316,756,303,634,461,458,718,861,951,455,543 72,410,899,157,214,610,812 49 256,923,577,521,058,878,088,611,477,224,235,621,321,607 506,876,294,100,502,275,687 51 12,589,255,298,531,885,026,341,962,383,987,545,444,758,743 3,548,134,058,703,515,929,815 53 616,873,509,628,062,366,290,756,156,815,389,726,793,178,407 24,836,938,410,924,611,508,707 55 30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943 173,858,568,876,472,280,560,953 57 1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207 1,217,009,982,135,305,963,926,677 59 72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143 8,519,069,874,947,141,747,486,745 61 3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007 59,633,489,124,629,992,232,407,216 63 174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343 417,434,423,872,409,945,626,850,517 65 8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807 2,922,040,967,106,869,619,387,953,625 67 418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543 20,454,286,769,748,087,335,715,675,381 69 20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607 143,180,007,388,236,611,350,009,727,669 71 1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 1,002,260,051,717,656,279,450,068,093,686 73 49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 7,015,820,362,023,593,956,150,476,655,802
jq[edit]
The following program takes advantage of the support for unbounded-precision
integer arithmetic provided by gojq, the Go implementation of jq, but it can also be run, with different numerical results, using the C implementation.# For gojq
def idivide($j):
. as $i
| ($i % $j) as $mod
| ($i - $mod) / $j ;
# input should be non-negative
def isqrt:
. as $x
| 1 | until(. > $x; . * 4) as $q
| {$q, $x, r: 0}
| until( .q <= 1;
.q |= idivide(4)
| .t = .x - .r - .q
| .r |= idivide(2)
| if .t >= 0
then .x = .t
| .r += .q
else . end).r ;
def power($n):
. as $in
| reduce range(0;$n) as $i (1; . * $in);
def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;
## The task:
"The integer square roots of integers from 0 to 65 are:",
[range(0;66) | isqrt],
"",
"The integer square roots of odd powers of 7 from 7^1 up to 7^73 are:",
("power" + " "*16 + "7 ^ power" + " "*70 + "integer square root"),
(range( 1;74;2) as $i
| (7 | power($i)) as $p
| "\($i|lpad(2)) \($p|lpad(84)) \($p | isqrt | lpad(43))" )
- Output:
Invocation: gojq -ncr -f isqrt.jq
The integer square roots of integers from 0 to 65 are: [0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8] The integer square roots of odd powers of 7 from 7^1 up to 7^73 are: power 7 ^ power integer square root 1 7 2 3 343 18 5 16807 129 7 823543 907 9 40353607 6352 11 1977326743 44467 13 96889010407 311269 15 4747561509943 2178889 17 232630513987207 15252229 19 11398895185373143 106765608 21 558545864083284007 747359260 23 27368747340080916343 5231514822 25 1341068619663964900807 36620603758 27 65712362363534280139543 256344226312 29 3219905755813179726837607 1794409584184 31 157775382034845806615042743 12560867089291 33 7730993719707444524137094407 87926069625040 35 378818692265664781682717625943 615482487375282 37 18562115921017574302453163671207 4308377411626977 39 909543680129861140820205019889143 30158641881388842 41 44567640326363195900190045974568007 211110493169721897 43 2183814375991796599109312252753832343 1477773452188053281 45 107006904423598033356356300384937784807 10344414165316372973 47 5243338316756303634461458718861951455543 72410899157214610812 49 256923577521058878088611477224235621321607 506876294100502275687 51 12589255298531885026341962383987545444758743 3548134058703515929815 53 616873509628062366290756156815389726793178407 24836938410924611508707 55 30226801971775055948247051683954096612865741943 173858568876472280560953 57 1481113296616977741464105532513750734030421355207 1217009982135305963926677 59 72574551534231909331741171093173785967490646405143 8519069874947141747486745 61 3556153025177363557255317383565515512407041673852007 59633489124629992232407216 63 174251498233690814305510551794710260107945042018748343 417434423872409945626850517 65 8538323413450849900970017037940802745289307058918668807 2922040967106869619387953625 67 418377847259091645147530834859099334519176045887014771543 20454286769748087335715675381 69 20500514515695490612229010908095867391439626248463723805607 143180007388236611350009727669 71 1004525211269079039999221534496697502180541686174722466474743 1002260051717656279450068093686 73 49221735352184872959961855190338177606846542622561400857262407 7015820362023593956150476655802
Julia[edit]
Julia also has a built in isqrt() function which works on integer types, but the function integer_sqrt is shown for the task.
using Formatting
function integer_sqrt(x)
@assert(x >= 0)
q = one(x)
while q <= x
q <<= 2
end
z, r = x, zero(x)
while q > 1
q >>= 2
t = z - r - q
r >>= 1
if t >= 0
z = t
r += q
end
end
return r
end
println("The integer square roots of integers from 0 to 65 are:")
println(integer_sqrt.(collect(0:65)))
println("\nThe integer square roots of odd powers of 7 from 7^1 up to 7^73 are:\n")
println("power", " "^36, "7 ^ power", " "^60, "integer square root")
println("----- ", "-"^80, " ------------------------------------------")
pow7 = big"7"
for i in 1:2:73
println(lpad(i, 2), lpad(format(pow7^i, commas=true), 84),
lpad(format(integer_sqrt(pow7^i), commas=true), 43))
end
- Output:
The integer square roots of integers from 0 to 65 are: [0, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8] The integer square roots of odd powers of 7 from 7^1 up to 7^73 are: power 7 ^ power integer square root ----- -------------------------------------------------------------------------------- ------------------------------------------ 1 7 2 3 343 18 5 16,807 129 7 823,543 907 9 40,353,607 6,352 11 1,977,326,743 44,467 13 96,889,010,407 311,269 15 4,747,561,509,943 2,178,889 17 232,630,513,987,207 15,252,229 19 11,398,895,185,373,143 106,765,608 21 558,545,864,083,284,007 747,359,260 23 27,368,747,340,080,916,343 5,231,514,822 25 1,341,068,619,663,964,900,807 36,620,603,758 27 65,712,362,363,534,280,139,543 256,344,226,312 29 3,219,905,755,813,179,726,837,607 1,794,409,584,184 31 157,775,382,034,845,806,615,042,743 12,560,867,089,291 33 7,730,993,719,707,444,524,137,094,407 87,926,069,625,040 35 378,818,692,265,664,781,682,717,625,943 615,482,487,375,282 37 18,562,115,921,017,574,302,453,163,671,207 4,308,377,411,626,977 39 909,543,680,129,861,140,820,205,019,889,143 30,158,641,881,388,842 41 44,567,640,326,363,195,900,190,045,974,568,007 211,110,493,169,721,897 43 2,183,814,375,991,796,599,109,312,252,753,832,343 1,477,773,452,188,053,281 45 107,006,904,423,598,033,356,356,300,384,937,784,807 10,344,414,165,316,372,973 47 5,243,338,316,756,303,634,461,458,718,861,951,455,543 72,410,899,157,214,610,812 49 256,923,577,521,058,878,088,611,477,224,235,621,321,607 506,876,294,100,502,275,687 51 12,589,255,298,531,885,026,341,962,383,987,545,444,758,743 3,548,134,058,703,515,929,815 53 616,873,509,628,062,366,290,756,156,815,389,726,793,178,407 24,836,938,410,924,611,508,707 55 30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943 173,858,568,876,472,280,560,953 57 1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207 1,217,009,982,135,305,963,926,677 59 72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143 8,519,069,874,947,141,747,486,745 61 3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007 59,633,489,124,629,992,232,407,216 63 174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343 417,434,423,872,409,945,626,850,517 65 8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807 2,922,040,967,106,869,619,387,953,625 67 418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543 20,454,286,769,748,087,335,715,675,381 69 20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607 143,180,007,388,236,611,350,009,727,669 71 1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 1,002,260,051,717,656,279,450,068,093,686 73 49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 7,015,820,362,023,593,956,150,476,655,802
Kotlin[edit]
import java.math.BigInteger
fun isqrt(x: BigInteger): BigInteger {
if (x < BigInteger.ZERO) {
throw IllegalArgumentException("Argument cannot be negative")
}
var q = BigInteger.ONE
while (q <= x) {
q = q.shiftLeft(2)
}
var z = x
var r = BigInteger.ZERO
while (q > BigInteger.ONE) {
q = q.shiftRight(2)
var t = z
t -= r
t -= q
r = r.shiftRight(1)
if (t >= BigInteger.ZERO) {
z = t
r += q
}
}
return r
}
fun main() {
println("The integer square root of integers from 0 to 65 are:")
for (i in 0..65) {
print("${isqrt(BigInteger.valueOf(i.toLong()))} ")
}
println()
println("The integer square roots of powers of 7 from 7^1 up to 7^73 are:")
println("power 7 ^ power integer square root")
println("----- --------------------------------------------------------------------------------- -----------------------------------------")
var pow7 = BigInteger.valueOf(7)
val bi49 = BigInteger.valueOf(49)
for (i in (1..73).step(2)) {
println("%2d %,84d %,41d".format(i, pow7, isqrt(pow7)))
pow7 *= bi49
}
}
- Output:
The integer square root of integers from 0 to 65 are: 0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 The integer square roots of powers of 7 from 7^1 up to 7^73 are: power 7 ^ power integer square root ----- --------------------------------------------------------------------------------- ----------------------------------------- 1 7 2 3 343 18 5 16,807 129 7 823,543 907 9 40,353,607 6,352 11 1,977,326,743 44,467 13 96,889,010,407 311,269 15 4,747,561,509,943 2,178,889 17 232,630,513,987,207 15,252,229 19 11,398,895,185,373,143 106,765,608 21 558,545,864,083,284,007 747,359,260 23 27,368,747,340,080,916,343 5,231,514,822 25 1,341,068,619,663,964,900,807 36,620,603,758 27 65,712,362,363,534,280,139,543 256,344,226,312 29 3,219,905,755,813,179,726,837,607 1,794,409,584,184 31 157,775,382,034,845,806,615,042,743 12,560,867,089,291 33 7,730,993,719,707,444,524,137,094,407 87,926,069,625,040 35 378,818,692,265,664,781,682,717,625,943 615,482,487,375,282 37 18,562,115,921,017,574,302,453,163,671,207 4,308,377,411,626,977 39 909,543,680,129,861,140,820,205,019,889,143 30,158,641,881,388,842 41 44,567,640,326,363,195,900,190,045,974,568,007 211,110,493,169,721,897 43 2,183,814,375,991,796,599,109,312,252,753,832,343 1,477,773,452,188,053,281 45 107,006,904,423,598,033,356,356,300,384,937,784,807 10,344,414,165,316,372,973 47 5,243,338,316,756,303,634,461,458,718,861,951,455,543 72,410,899,157,214,610,812 49 256,923,577,521,058,878,088,611,477,224,235,621,321,607 506,876,294,100,502,275,687 51 12,589,255,298,531,885,026,341,962,383,987,545,444,758,743 3,548,134,058,703,515,929,815 53 616,873,509,628,062,366,290,756,156,815,389,726,793,178,407 24,836,938,410,924,611,508,707 55 30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943 173,858,568,876,472,280,560,953 57 1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207 1,217,009,982,135,305,963,926,677 59 72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143 8,519,069,874,947,141,747,486,745 61 3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007 59,633,489,124,629,992,232,407,216 63 174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343 417,434,423,872,409,945,626,850,517 65 8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807 2,922,040,967,106,869,619,387,953,625 67 418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543 20,454,286,769,748,087,335,715,675,381 69 20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607 143,180,007,388,236,611,350,009,727,669 71 1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 1,002,260,051,717,656,279,450,068,093,686 73 49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 7,015,820,362,023,593,956,150,476,655,802
Lua[edit]
function isqrt(x)
local q = 1
local r = 0
while q <= x do
q = q << 2
end
while q > 1 do
q = q >> 2
local t = x - r - q
r = r >> 1
if t >= 0 then
x = t
r = r + q
end
end
return r
end
print("Integer square root for numbers 0 to 65:")
for n=0,65 do
io.write(isqrt(n) .. ' ')
end
print()
print()
print("Integer square roots of oddd powers of 7 from 1 to 21:")
print(" n | 7 ^ n | isqrt(7 ^ n)")
local p = 7
local n = 1
while n <= 21 do
print(string.format("%2d | %18d | %12d", n, p, isqrt(p)))
----------------------
n = n + 2
p = p * 49
end
- Output:
Integer square root for numbers 0 to 65: 0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 Integer square roots of oddd powers of 7 from 1 to 21: n | 7 ^ n | isqrt(7 ^ n) 3 | 343 | 18 5 | 16807 | 129 7 | 823543 | 907 9 | 40353607 | 6352 11 | 1977326743 | 44467 13 | 96889010407 | 311269 15 | 4747561509943 | 2178889 17 | 232630513987207 | 15252229 19 | 11398895185373143 | 106765608 21 | 558545864083284007 | 747359260
MAD[edit]
NORMAL MODE IS INTEGER
R INTEGER SQUARE ROOT OF X
INTERNAL FUNCTION(X)
ENTRY TO ISQRT.
Q = 1
FNDPW4 WHENEVER Q.LE.X
Q = Q * 4
TRANSFER TO FNDPW4
END OF CONDITIONAL
Z = X
R = 0
FNDRT WHENEVER Q.G.1
Q = Q / 4
T = Z - R - Q
R = R / 2
WHENEVER T.GE.0
Z = T
R = R + Q
END OF CONDITIONAL
TRANSFER TO FNDRT
END OF CONDITIONAL
FUNCTION RETURN R
END OF FUNCTION
R PRINT INTEGER SQUARE ROOTS OF 0..65
THROUGH SQ65, FOR N=0, 11, N.G.65
SQ65 PRINT FORMAT N11, ISQRT.(N), ISQRT.(N+1), ISQRT.(N+2),
0 ISQRT.(N+3), ISQRT.(N+4), ISQRT.(N+5), ISQRT.(N+6),
1 ISQRT.(N+7), ISQRT.(N+8), ISQRT.(N+9), ISQRT.(N+10)
VECTOR VALUES N11 = $11(I1,S1)*$
R MACHINE WORD SIZE ON IBM 704 IS 36 BITS
R PRINT UP TO AND INCLUDING ISQRT(7^12)
POW7 = 1
THROUGH SQ7P12, FOR I=0, 1, I.G.12
PRINT FORMAT SQ7, I, ISQRT.(POW7)
SQ7P12 POW7 = POW7 * 7
VECTOR VALUES SQ7 = $9HISQRT.(7^,I2,4H) = ,I6*$
END OF PROGRAM
- Output:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 ISQRT.(7^ 0) = 1 ISQRT.(7^ 1) = 2 ISQRT.(7^ 2) = 7 ISQRT.(7^ 3) = 18 ISQRT.(7^ 4) = 49 ISQRT.(7^ 5) = 129 ISQRT.(7^ 6) = 343 ISQRT.(7^ 7) = 907 ISQRT.(7^ 8) = 2401 ISQRT.(7^ 9) = 6352 ISQRT.(7^10) = 16807 ISQRT.(7^11) = 44467 ISQRT.(7^12) = 117649
Mathematica/Wolfram Language[edit]
ClearAll[ISqrt]
ISqrt[x_Integer?NonNegative] := Module[{q = 1, z, r, t},
While[q <= x,
q *= 4
];
z = x;
r = 0;
While[q > 1,
q = Quotient[q, 4];
t = z - r - q;
r /= 2;
If[t >= 0,
z = t;
r += q
];
];
r
]
ISqrt /@ Range[65]
Column[ISqrt /@ (7^Range[1, 73])]
- Output:
{1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8}
2
7
18
49
129
343
907
2401
6352
16807
44467
117649
311269
823543
2178889
5764801
15252229
40353607
106765608
282475249
747359260
1977326743
5231514822
13841287201
36620603758
96889010407
256344226312
678223072849
1794409584184
4747561509943
12560867089291
33232930569601
87926069625040
232630513987207
615482487375282
1628413597910449
4308377411626977
11398895185373143
30158641881388842
79792266297612001
211110493169721897
558545864083284007
1477773452188053281
3909821048582988049
10344414165316372973
27368747340080916343
72410899157214610812
191581231380566414401
506876294100502275687
1341068619663964900807
3548134058703515929815
9387480337647754305649
24836938410924611508707
65712362363534280139543
173858568876472280560953
459986536544739960976801
1217009982135305963926677
3219905755813179726837607
8519069874947141747486745
22539340290692258087863249
59633489124629992232407216
157775382034845806615042743
417434423872409945626850517
1104427674243920646305299201
2922040967106869619387953625
7730993719707444524137094407
20454286769748087335715675381
54116956037952111668959660849
143180007388236611350009727669
378818692265664781682717625943
1002260051717656279450068093686
2651730845859653471779023381601
7015820362023593956150476655802
Nim[edit]
This Nim implementation provides an isqrt function for signed integers and for big integers.
import strformat, strutils
import bignum
func isqrt*[T: SomeSignedInt | Int](x: T): T =
## Compute integer square root for signed integers
## and for big integers.
when T is Int:
result = newInt()
var q = newInt(1)
else:
result = 0
var q = T(1)
while q <= x:
q = q shl 2
var z = x
while q > 1:
q = q shr 2
let t = z - result - q
result = result shr 1
if t >= 0:
z = t
result += q
when isMainModule:
echo "Integer square root for numbers 0 to 65:"
for n in 0..65:
stdout.write ' ', isqrt(n)
echo "\n"
echo "Integer square roots of odd powers of 7 from 7^1 to 7^73:"
echo " n" & repeat(' ', 82) & "7^n" & repeat(' ', 34) & "isqrt(7^n)"
echo repeat("—", 131)
var x = newInt(7)
for n in countup(1, 73, 2):
echo &"{n:>2} {insertSep($x, ','):>82} {insertSep($isqrt(x), ','):>41}"
x *= 49
- Output:
Integer square root for numbers 0 to 65: 0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 Integer square roots of odd powers of 7 from 7^1 to 7^73: n 7^n isqrt(7^n) ——————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————— 1 7 2 3 343 18 5 16,807 129 7 823,543 907 9 40,353,607 6,352 11 1,977,326,743 44,467 13 96,889,010,407 311,269 15 4,747,561,509,943 2,178,889 17 232,630,513,987,207 15,252,229 19 11,398,895,185,373,143 106,765,608 21 558,545,864,083,284,007 747,359,260 23 27,368,747,340,080,916,343 5,231,514,822 25 1,341,068,619,663,964,900,807 36,620,603,758 27 65,712,362,363,534,280,139,543 256,344,226,312 29 3,219,905,755,813,179,726,837,607 1,794,409,584,184 31 157,775,382,034,845,806,615,042,743 12,560,867,089,291 33 7,730,993,719,707,444,524,137,094,407 87,926,069,625,040 35 378,818,692,265,664,781,682,717,625,943 615,482,487,375,282 37 18,562,115,921,017,574,302,453,163,671,207 4,308,377,411,626,977 39 909,543,680,129,861,140,820,205,019,889,143 30,158,641,881,388,842 41 44,567,640,326,363,195,900,190,045,974,568,007 211,110,493,169,721,897 43 2,183,814,375,991,796,599,109,312,252,753,832,343 1,477,773,452,188,053,281 45 107,006,904,423,598,033,356,356,300,384,937,784,807 10,344,414,165,316,372,973 47 5,243,338,316,756,303,634,461,458,718,861,951,455,543 72,410,899,157,214,610,812 49 256,923,577,521,058,878,088,611,477,224,235,621,321,607 506,876,294,100,502,275,687 51 12,589,255,298,531,885,026,341,962,383,987,545,444,758,743 3,548,134,058,703,515,929,815 53 616,873,509,628,062,366,290,756,156,815,389,726,793,178,407 24,836,938,410,924,611,508,707 55 30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943 173,858,568,876,472,280,560,953 57 1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207 1,217,009,982,135,305,963,926,677 59 72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143 8,519,069,874,947,141,747,486,745 61 3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007 59,633,489,124,629,992,232,407,216 63 174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343 417,434,423,872,409,945,626,850,517 65 8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807 2,922,040,967,106,869,619,387,953,625 67 418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543 20,454,286,769,748,087,335,715,675,381 69 20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607 143,180,007,388,236,611,350,009,727,669 71 1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 1,002,260,051,717,656,279,450,068,093,686 73 49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 7,015,820,362,023,593,956,150,476,655,802
Ol[edit]
(print "Integer square roots of 0..65")
(for-each (lambda (x)
(display (isqrt x))
(display " "))
(iota 66))
(print)
(print "Integer square roots of 7^n")
(for-each (lambda (x)
(print "x: " x ", isqrt: " (isqrt x)))
(map (lambda (i)
(expt 7 i))
(iota 73 1)))
(print)
- Output:
Integer square roots of 0..65 0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 Integer square roots of 7^n x: 7, isqrt: 2 x: 49, isqrt: 7 x: 343, isqrt: 18 x: 2401, isqrt: 49 x: 16807, isqrt: 129 x: 117649, isqrt: 343 x: 823543, isqrt: 907 x: 5764801, isqrt: 2401 x: 40353607, isqrt: 6352 x: 282475249, isqrt: 16807 x: 1977326743, isqrt: 44467 x: 13841287201, isqrt: 117649 x: 96889010407, isqrt: 311269 x: 678223072849, isqrt: 823543 x: 4747561509943, isqrt: 2178889 x: 33232930569601, isqrt: 5764801 x: 232630513987207, isqrt: 15252229 x: 1628413597910449, isqrt: 40353607 x: 11398895185373143, isqrt: 106765608 x: 79792266297612001, isqrt: 282475249 x: 558545864083284007, isqrt: 747359260 x: 3909821048582988049, isqrt: 1977326743 x: 27368747340080916343, isqrt: 5231514822 x: 191581231380566414401, isqrt: 13841287201 x: 1341068619663964900807, isqrt: 36620603758 x: 9387480337647754305649, isqrt: 96889010407 x: 65712362363534280139543, isqrt: 256344226312 x: 459986536544739960976801, isqrt: 678223072849 x: 3219905755813179726837607, isqrt: 1794409584184 x: 22539340290692258087863249, isqrt: 4747561509943 x: 157775382034845806615042743, isqrt: 12560867089291 x: 1104427674243920646305299201, isqrt: 33232930569601 x: 7730993719707444524137094407, isqrt: 87926069625040 x: 54116956037952111668959660849, isqrt: 232630513987207 x: 378818692265664781682717625943, isqrt: 615482487375282 x: 2651730845859653471779023381601, isqrt: 1628413597910449 x: 18562115921017574302453163671207, isqrt: 4308377411626977 x: 129934811447123020117172145698449, isqrt: 11398895185373143 x: 909543680129861140820205019889143, isqrt: 30158641881388842 x: 6366805760909027985741435139224001, isqrt: 79792266297612001 x: 44567640326363195900190045974568007, isqrt: 211110493169721897 x: 311973482284542371301330321821976049, isqrt: 558545864083284007 x: 2183814375991796599109312252753832343, isqrt: 1477773452188053281 x: 15286700631942576193765185769276826401, isqrt: 3909821048582988049 x: 107006904423598033356356300384937784807, isqrt: 10344414165316372973 x: 749048330965186233494494102694564493649, isqrt: 27368747340080916343 x: 5243338316756303634461458718861951455543, isqrt: 72410899157214610812 x: 36703368217294125441230211032033660188801, isqrt: 191581231380566414401 x: 256923577521058878088611477224235621321607, isqrt: 506876294100502275687 x: 1798465042647412146620280340569649349251249, isqrt: 1341068619663964900807 x: 12589255298531885026341962383987545444758743, isqrt: 3548134058703515929815 x: 88124787089723195184393736687912818113311201, isqrt: 9387480337647754305649 x: 616873509628062366290756156815389726793178407, isqrt: 24836938410924611508707 x: 4318114567396436564035293097707728087552248849, isqrt: 65712362363534280139543 x: 30226801971775055948247051683954096612865741943, isqrt: 173858568876472280560953 x: 211587613802425391637729361787678676290060193601, isqrt: 459986536544739960976801 x: 1481113296616977741464105532513750734030421355207, isqrt: 1217009982135305963926677 x: 10367793076318844190248738727596255138212949486449, isqrt: 3219905755813179726837607 x: 72574551534231909331741171093173785967490646405143, isqrt: 8519069874947141747486745 x: 508021860739623365322188197652216501772434524836001, isqrt: 22539340290692258087863249 x: 3556153025177363557255317383565515512407041673852007, isqrt: 59633489124629992232407216 x: 24893071176241544900787221684958608586849291716964049, isqrt: 157775382034845806615042743 x: 174251498233690814305510551794710260107945042018748343, isqrt: 417434423872409945626850517 x: 1219760487635835700138573862562971820755615294131238401, isqrt: 1104427674243920646305299201 x: 8538323413450849900970017037940802745289307058918668807, isqrt: 2922040967106869619387953625 x: 59768263894155949306790119265585619217025149412430681649, isqrt: 7730993719707444524137094407 x: 418377847259091645147530834859099334519176045887014771543, isqrt: 20454286769748087335715675381 x: 2928644930813641516032715844013695341634232321209103400801, isqrt: 54116956037952111668959660849 x: 20500514515695490612229010908095867391439626248463723805607, isqrt: 143180007388236611350009727669 x: 143503601609868434285603076356671071740077383739246066639249, isqrt: 378818692265664781682717625943 x: 1004525211269079039999221534496697502180541686174722466474743, isqrt: 1002260051717656279450068093686 x: 7031676478883553279994550741476882515263791803223057265323201, isqrt: 2651730845859653471779023381601 x: 49221735352184872959961855190338177606846542622561400857262407, isqrt: 7015820362023593956150476655802
Pascal[edit]
[1]
//************************************************//
// //
// Thanks for rvelthuis for BigIntegers library //
// https://github.com/rvelthuis/DelphiBigNumbers //
// //
//************************************************//
program IsqrtTask;
{$APPTYPE CONSOLE}
{$R *.res}
uses
System.SysUtils,
Velthuis.BigIntegers;
function isqrt(x: BigInteger): BigInteger;
var
q, r, t: BigInteger;
begin
q := 1;
r := 0;
while (q <= x) do
q := q shl 2;
while (q > 1) do
begin
q := q shr 2;
t := x - r - q;
r := r shr 1;
if (t >= 0) then
begin
x := t;
r := r + q;
end;
end;
Result := r;
end;
function commatize(const n: BigInteger; size: Integer): string;
var
str: string;
digits: Integer;
i: Integer;
begin
Result := '';
str := n.ToString;
digits := str.Length;
for i := 1 to digits do
begin
if ((i > 1) and (((i - 1) mod 3) = (digits mod 3))) then
Result := Result + ',';
Result := Result + str[i];
end;
if Result.Length < size then
Result := string.Create(' ', size - Result.Length) + Result;
end;
const
POWER_WIDTH = 83;
ISQRT_WIDTH = 42;
var
n, i: Integer;
f: TextFile;
p: BigInteger;
begin
AssignFile(f, 'output.txt');
rewrite(f);
Writeln(f, 'Integer square root for numbers 0 to 65:');
for n := 0 to 65 do
Write(f, isqrt(n).ToString, ' ');
Writeln(f, #10#10'Integer square roots of odd powers of 7 from 1 to 73:');
Write(f, ' n |', string.Create(' ', 78), '7 ^ n |', string.Create(' ', 30),
'isqrt(7 ^ n)'#10);
Writeln(f, string.Create('-', 17 + POWER_WIDTH + ISQRT_WIDTH));
p := 7;
n := 1;
repeat
Writeln(f, Format('%2d', [n]), ' |', commatize(p, power_width), ' |',
commatize(isqrt(p), isqrt_width));
inc(n, 2);
p := p * 49;
until (n > 73);
CloseFile(f);
end.
Perl[edit]
# 20201029 added Perl programming solution
use strict;
use warnings;
use bigint;
use CLDR::Number 'decimal_formatter';
sub integer_sqrt {
( my $x = $_[0] ) >= 0 or die;
my $q = 1;
while ($q <= $x) {
$q <<= 2
}
my ($z, $r) = ($x, 0);
while ($q > 1) {
$q >>= 2;
my $t = $z - $r - $q;
$r >>= 1;
if ($t >= 0) {
$z = $t;
$r += $q;
}
}
return $r
}
print "The integer square roots of integers from 0 to 65 are:\n";
print map { ( integer_sqrt $_ ) . ' ' } (0..65);
my $cldr = CLDR::Number->new();
my $decf = $cldr->decimal_formatter;
print "\nThe integer square roots of odd powers of 7 from 7^1 up to 7^73 are:\n";
print "power", " "x36, "7 ^ power", " "x60, "integer square root\n";
print "----- ", "-"x79, " ------------------------------------------\n";
for (my $i = 1; $i < 74; $i += 2) {
printf("%2s ", $i);
printf("%82s", $decf->format( 7**$i ) );
printf("%44s", $decf->format( integer_sqrt(7**$i) ) ) ;
print "\n";
}
- Output:
The integer square roots of integers from 0 to 65 are: 0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 The integer square roots of odd powers of 7 from 7^1 up to 7^73 are: power 7 ^ power integer square root ----- ------------------------------------------------------------------------------- ------------------------------------------ 1 7 2 3 343 18 5 16,807 129 7 823,543 907 9 40,353,607 6,352 11 1,977,326,743 44,467 13 96,889,010,407 311,269 15 4,747,561,509,943 2,178,889 17 232,630,513,987,207 15,252,229 19 11,398,895,185,373,143 106,765,608 21 558,545,864,083,284,007 747,359,260 23 27,368,747,340,080,916,343 5,231,514,822 25 1,341,068,619,663,964,900,807 36,620,603,758 27 65,712,362,363,534,280,139,543 256,344,226,312 29 3,219,905,755,813,179,726,837,607 1,794,409,584,184 31 157,775,382,034,845,806,615,042,743 12,560,867,089,291 33 7,730,993,719,707,444,524,137,094,407 87,926,069,625,040 35 378,818,692,265,664,781,682,717,625,943 615,482,487,375,282 37 18,562,115,921,017,574,302,453,163,671,207 4,308,377,411,626,977 39 909,543,680,129,861,140,820,205,019,889,143 30,158,641,881,388,842 41 44,567,640,326,363,195,900,190,045,974,568,007 211,110,493,169,721,897 43 2,183,814,375,991,796,599,109,312,252,753,832,343 1,477,773,452,188,053,281 45 107,006,904,423,598,033,356,356,300,384,937,784,807 10,344,414,165,316,372,973 47 5,243,338,316,756,303,634,461,458,718,861,951,455,543 72,410,899,157,214,610,812 49 256,923,577,521,058,878,088,611,477,224,235,621,321,607 506,876,294,100,502,275,687 51 12,589,255,298,531,885,026,341,962,383,987,545,444,758,743 3,548,134,058,703,515,929,815 53 616,873,509,628,062,366,290,756,156,815,389,726,793,178,407 24,836,938,410,924,611,508,707 55 30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943 173,858,568,876,472,280,560,953 57 1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207 1,217,009,982,135,305,963,926,677 59 72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143 8,519,069,874,947,141,747,486,745 61 3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007 59,633,489,124,629,992,232,407,216 63 174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343 417,434,423,872,409,945,626,850,517 65 8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807 2,922,040,967,106,869,619,387,953,625 67 418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543 20,454,286,769,748,087,335,715,675,381 69 20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607 143,180,007,388,236,611,350,009,727,669 71 1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 1,002,260,051,717,656,279,450,068,093,686 73 49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 7,015,820,362,023,593,956,150,476,655,802
Phix[edit]
See also Integer_roots#Phix for a simpler and shorter example using the mpz_root() routine, or better yet just use mpz_root() directly (that is, rather than the isqrt() below).
with javascript_semantics include mpfr.e function isqrt(mpz x) if mpz_cmp_si(x,0)<0 then crash("Argument cannot be negative.") end if mpz q = mpz_init(1), r = mpz_init(0), t = mpz_init(), z = mpz_init_set(x) while mpz_cmp(q,x)<= 0 do mpz_mul_si(q,q,4) end while while mpz_cmp_si(q,1)>0 do assert(mpz_fdiv_q_ui(q, q, 4)=0) mpz_sub(t,z,r) mpz_sub(t,t,q) assert(mpz_fdiv_q_ui(r, r, 2)=0) if mpz_cmp_si(t,0) >= 0 then mpz_set(z,t) mpz_add(r,r,q) end if end while string star = iff(mpz_cmp_si(z,0)=0?"*":" ") return shorten(mpz_get_str(r,10,true))&star end function printf(1,"The integer square roots of integers from 0 to 65 are:\n") for i=0 to 65 do printf(1,"%s ", {trim(isqrt(mpz_init(i)))}) end for printf(1,"\n\npower 7 ^ power integer square root\n") printf(1,"----- --------------------------------------------------------- ----------------------------------------------------------\n") mpz pow7 = mpz_init(7) for i=1 to 9000 do if (i<=73 and remainder(i,2)=1) or (i<100 and remainder(i,10)=5) or (i<1000 and remainder(i,100)=0) or remainder(i,1000)=0 then printf(1,"%4d %58s %61s\n", {i, shorten(mpz_get_str(pow7,10,true)),isqrt(pow7)}) end if mpz_mul_si(pow7,pow7,7) end for
- Output:
Perfect squares are denoted with an asterisk.
The integer square roots of integers from 0 to 65 are: 0* 1* 1 1 2* 2 2 2 2 3* 3 3 3 3 3 3 4* 4 4 4 4 4 4 4 4 5* 5 5 5 5 5 5 5 5 5 5 6* 6 6 6 6 6 6 6 6 6 6 6 6 7* 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8* 8 power 7 ^ power integer square root ----- --------------------------------------------------------- ---------------------------------------------------------- 1 7 2 3 343 18 5 16,807 129 7 823,543 907 9 40,353,607 6,352 11 1,977,326,743 44,467 13 96,889,010,407 311,269 15 4,747,561,509,943 2,178,889 17 232,630,513,987,207 15,252,229 19 11,398,895,185,373,143 106,765,608 21 558,545,864,083,284,007 747,359,260 23 27,368,747,340,080,916,343 5,231,514,822 25 1,341,068,619,663,964,900,807 36,620,603,758 27 65,712,362,363,534,280,139,543 256,344,226,312 29 3,219,905,755,813,179,726,837,607 1,794,409,584,184 31 157,775,382,034,845,806,615,042,743 12,560,867,089,291 33 7,730,993,719,707,444,524,137,094,407 87,926,069,625,040 35 378,818,692,265,664,781,682,717,625,943 615,482,487,375,282 37 18,562,115,921,017,574,302,453,163,671,207 4,308,377,411,626,977 39 909,543,680,129,861,140,820,205,019,889,143 30,158,641,881,388,842 41 44,567,640,326,363,195,900,190,045,974,568,007 211,110,493,169,721,897 43 2,183,814,375,991,796,599,109,312,252,753,832,343 1,477,773,452,188,053,281 45 107,006,904,423,598,033,356,356,300,384,937,784,807 10,344,414,165,316,372,973 47 5,243,338,316,756,303,634,461,458,718,861,951,455,543 72,410,899,157,214,610,812 49 256,923,577,521,058,878,088,611,477,224,235,621,321,607 506,876,294,100,502,275,687 51 12,589,255,298,531,8...,987,545,444,758,743 (44 digits) 3,548,134,058,703,515,929,815 53 616,873,509,628,062,...,389,726,793,178,407 (45 digits) 24,836,938,410,924,611,508,707 55 30,226,801,971,775,0...,096,612,865,741,943 (47 digits) 173,858,568,876,472,280,560,953 57 1,481,113,296,616,97...,734,030,421,355,207 (49 digits) 1,217,009,982,135,305,963,926,677 59 72,574,551,534,231,9...,967,490,646,405,143 (50 digits) 8,519,069,874,947,141,747,486,745 61 3,556,153,025,177,36...,407,041,673,852,007 (52 digits) 59,633,489,124,629,992,232,407,216 63 174,251,498,233,690,...,945,042,018,748,343 (54 digits) 417,434,423,872,409,945,626,850,517 65 8,538,323,413,450,84...,307,058,918,668,807 (55 digits) 2,922,040,967,106,869,619,387,953,625 67 418,377,847,259,091,...,045,887,014,771,543 (57 digits) 20,454,286,769,748,087,335,715,675,381 69 20,500,514,515,695,4...,248,463,723,805,607 (59 digits) 143,180,007,388,236,611,350,009,727,669 71 1,004,525,211,269,07...,174,722,466,474,743 (61 digits) 1,002,260,051,717,656,279,450,068,093,686 73 49,221,735,352,184,8...,561,400,857,262,407 (62 digits) 7,015,820,362,023,593,956,150,476,655,802 75 2,411,865,032,257,05...,508,642,005,857,943 (64 digits) 49,110,742,534,165,157,693,053,336,590,618 85 681,292,175,541,205,...,256,581,907,552,807 (72 digits) 825,404,249,771,713,805,347,147,428,078,522,216 95 192,448,176,927,753,...,224,874,137,973,943 (81 digits) 13,872,569,225,913,193,926,469,506,823,715,722,892,042 100 3,234,476,509,624,75...,459,636,928,060,001 (85 digits) 1,798,465,042,647,41...,569,649,349,251,249 (43 digits)* 200 10,461,838,291,314,3...,534,637,456,120,001 (170 digits) 3,234,476,509,624,75...,459,636,928,060,001 (85 digits)* 300 33,838,570,200,749,1...,841,001,584,180,001 (254 digits) 5,817,092,933,824,34...,721,127,496,191,249 (127 digits)* 400 109,450,060,433,611,...,994,729,312,240,001 (339 digits) 10,461,838,291,314,3...,534,637,456,120,001 (170 digits)* 500 354,013,649,449,525,...,611,820,640,300,001 (423 digits) 18,815,250,448,759,0...,761,742,043,131,249 (212 digits)* 600 1,145,048,833,231,02...,308,275,568,360,001 (508 digits) 33,838,570,200,749,1...,841,001,584,180,001 (254 digits)* 700 3,703,633,553,458,98...,700,094,096,420,001 (592 digits) 60,857,485,599,217,6...,075,492,990,071,249 (296 digits)* 800 11,979,315,728,921,1...,403,276,224,480,001 (677 digits) 109,450,060,433,611,...,994,729,312,240,001 (339 digits)* 900 38,746,815,326,573,9...,033,821,952,540,001 (761 digits) 196,842,107,605,496,...,046,380,337,011,249 (381 digits)* 1000 125,325,663,996,571,...,207,731,280,600,001 (846 digits) 354,013,649,449,525,...,611,820,640,300,001 (423 digits)* 2000 15,706,522,056,181,6...,351,822,561,200,001 (1,691 digits) 125,325,663,996,571,...,207,731,280,600,001 (846 digits)* 3000 1,968,430,305,767,76...,432,273,841,800,001 (2,536 digits) 44,366,995,681,111,4...,787,731,920,900,001 (1,268 digits)* 4000 246,694,835,101,319,...,449,085,122,400,001 (3,381 digits) 15,706,522,056,181,6...,351,822,561,200,001 (1,691 digits)* 5000 30,917,194,013,597,6...,402,256,403,000,001 (4,226 digits) 5,560,323,193,268,32...,900,003,201,500,001 (2,113 digits)* 6000 3,874,717,868,664,96...,291,787,683,600,001 (5,071 digits) 1,968,430,305,767,76...,432,273,841,800,001 (2,536 digits)* 7000 485,601,589,689,818,...,117,678,964,200,001 (5,916 digits) 696,851,196,231,891,...,948,634,482,100,001 (2,958 digits)* 8000 60,858,341,665,667,3...,879,930,244,800,001 (6,761 digits) 246,694,835,101,319,...,449,085,122,400,001 (3,381 digits)* 9000 7,627,112,078,979,99...,578,541,525,400,001 (7,606 digits) 87,333,338,874,567,2...,933,625,762,700,001 (3,803 digits)*
(Note that pre-0.8.2 the "(NNN digits)" count includes commas)
Python[edit]
def isqrt ( x ):
q = 1
while q <= x :
q *= 4
z,r = x,0
while q > 1 :
q /= 4
t,r = z-r-q,r/2
if t >= 0 :
z,r = t,r+q
return r
print ' '.join( '%d'%isqrt( n ) for n in xrange( 66 ))
print '\n'.join( '{0:114,} = isqrt( 7^{1:3} )'.format( isqrt( 7**n ),n ) for n in range( 1,204,2 ))
- Output:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8
2 = isqrt( 7^ 1 )
18 = isqrt( 7^ 3 )
129 = isqrt( 7^ 5 )
907 = isqrt( 7^ 7 )
6,352 = isqrt( 7^ 9 )
44,467 = isqrt( 7^ 11 )
311,269 = isqrt( 7^ 13 )
2,178,889 = isqrt( 7^ 15 )
15,252,229 = isqrt( 7^ 17 )
106,765,608 = isqrt( 7^ 19 )
747,359,260 = isqrt( 7^ 21 )
5,231,514,822 = isqrt( 7^ 23 )
36,620,603,758 = isqrt( 7^ 25 )
256,344,226,312 = isqrt( 7^ 27 )
1,794,409,584,184 = isqrt( 7^ 29 )
12,560,867,089,291 = isqrt( 7^ 31 )
87,926,069,625,040 = isqrt( 7^ 33 )
615,482,487,375,282 = isqrt( 7^ 35 )
4,308,377,411,626,977 = isqrt( 7^ 37 )
30,158,641,881,388,842 = isqrt( 7^ 39 )
211,110,493,169,721,897 = isqrt( 7^ 41 )
1,477,773,452,188,053,281 = isqrt( 7^ 43 )
10,344,414,165,316,372,973 = isqrt( 7^ 45 )
72,410,899,157,214,610,812 = isqrt( 7^ 47 )
506,876,294,100,502,275,687 = isqrt( 7^ 49 )
3,548,134,058,703,515,929,815 = isqrt( 7^ 51 )
24,836,938,410,924,611,508,707 = isqrt( 7^ 53 )
173,858,568,876,472,280,560,953 = isqrt( 7^ 55 )
1,217,009,982,135,305,963,926,677 = isqrt( 7^ 57 )
8,519,069,874,947,141,747,486,745 = isqrt( 7^ 59 )
59,633,489,124,629,992,232,407,216 = isqrt( 7^ 61 )
417,434,423,872,409,945,626,850,517 = isqrt( 7^ 63 )
2,922,040,967,106,869,619,387,953,625 = isqrt( 7^ 65 )
20,454,286,769,748,087,335,715,675,381 = isqrt( 7^ 67 )
143,180,007,388,236,611,350,009,727,669 = isqrt( 7^ 69 )
1,002,260,051,717,656,279,450,068,093,686 = isqrt( 7^ 71 )
7,015,820,362,023,593,956,150,476,655,802 = isqrt( 7^ 73 )
49,110,742,534,165,157,693,053,336,590,618 = isqrt( 7^ 75 )
343,775,197,739,156,103,851,373,356,134,328 = isqrt( 7^ 77 )
2,406,426,384,174,092,726,959,613,492,940,298 = isqrt( 7^ 79 )
16,844,984,689,218,649,088,717,294,450,582,086 = isqrt( 7^ 81 )
117,914,892,824,530,543,621,021,061,154,074,602 = isqrt( 7^ 83 )
825,404,249,771,713,805,347,147,428,078,522,216 = isqrt( 7^ 85 )
5,777,829,748,401,996,637,430,031,996,549,655,515 = isqrt( 7^ 87 )
40,444,808,238,813,976,462,010,223,975,847,588,606 = isqrt( 7^ 89 )
283,113,657,671,697,835,234,071,567,830,933,120,245 = isqrt( 7^ 91 )
1,981,795,603,701,884,846,638,500,974,816,531,841,720 = isqrt( 7^ 93 )
13,872,569,225,913,193,926,469,506,823,715,722,892,042 = isqrt( 7^ 95 )
97,107,984,581,392,357,485,286,547,766,010,060,244,299 = isqrt( 7^ 97 )
679,755,892,069,746,502,397,005,834,362,070,421,710,095 = isqrt( 7^ 99 )
4,758,291,244,488,225,516,779,040,840,534,492,951,970,665 = isqrt( 7^101 )
33,308,038,711,417,578,617,453,285,883,741,450,663,794,661 = isqrt( 7^103 )
233,156,270,979,923,050,322,173,001,186,190,154,646,562,631 = isqrt( 7^105 )
1,632,093,896,859,461,352,255,211,008,303,331,082,525,938,421 = isqrt( 7^107 )
11,424,657,278,016,229,465,786,477,058,123,317,577,681,568,950 = isqrt( 7^109 )
79,972,600,946,113,606,260,505,339,406,863,223,043,770,982,651 = isqrt( 7^111 )
559,808,206,622,795,243,823,537,375,848,042,561,306,396,878,562 = isqrt( 7^113 )
3,918,657,446,359,566,706,764,761,630,936,297,929,144,778,149,940 = isqrt( 7^115 )
27,430,602,124,516,966,947,353,331,416,554,085,504,013,447,049,581 = isqrt( 7^117 )
192,014,214,871,618,768,631,473,319,915,878,598,528,094,129,347,071 = isqrt( 7^119 )
1,344,099,504,101,331,380,420,313,239,411,150,189,696,658,905,429,502 = isqrt( 7^121 )
9,408,696,528,709,319,662,942,192,675,878,051,327,876,612,338,006,515 = isqrt( 7^123 )
65,860,875,700,965,237,640,595,348,731,146,359,295,136,286,366,045,605 = isqrt( 7^125 )
461,026,129,906,756,663,484,167,441,118,024,515,065,954,004,562,319,241 = isqrt( 7^127 )
3,227,182,909,347,296,644,389,172,087,826,171,605,461,678,031,936,234,687 = isqrt( 7^129 )
22,590,280,365,431,076,510,724,204,614,783,201,238,231,746,223,553,642,811 = isqrt( 7^131 )
158,131,962,558,017,535,575,069,432,303,482,408,667,622,223,564,875,499,679 = isqrt( 7^133 )
1,106,923,737,906,122,749,025,486,026,124,376,860,673,355,564,954,128,497,756 = isqrt( 7^135 )
7,748,466,165,342,859,243,178,402,182,870,638,024,713,488,954,678,899,484,295 = isqrt( 7^137 )
54,239,263,157,400,014,702,248,815,280,094,466,172,994,422,682,752,296,390,067 = isqrt( 7^139 )
379,674,842,101,800,102,915,741,706,960,661,263,210,960,958,779,266,074,730,470 = isqrt( 7^141 )
2,657,723,894,712,600,720,410,191,948,724,628,842,476,726,711,454,862,523,113,293 = isqrt( 7^143 )
18,604,067,262,988,205,042,871,343,641,072,401,897,337,086,980,184,037,661,793,056 = isqrt( 7^145 )
130,228,470,840,917,435,300,099,405,487,506,813,281,359,608,861,288,263,632,551,397 = isqrt( 7^147 )
911,599,295,886,422,047,100,695,838,412,547,692,969,517,262,029,017,845,427,859,782 = isqrt( 7^149 )
6,381,195,071,204,954,329,704,870,868,887,833,850,786,620,834,203,124,917,995,018,479 = isqrt( 7^151 )
44,668,365,498,434,680,307,934,096,082,214,836,955,506,345,839,421,874,425,965,129,358 = isqrt( 7^153 )
312,678,558,489,042,762,155,538,672,575,503,858,688,544,420,875,953,120,981,755,905,510 = isqrt( 7^155 )
2,188,749,909,423,299,335,088,770,708,028,527,010,819,810,946,131,671,846,872,291,338,571 = isqrt( 7^157 )
15,321,249,365,963,095,345,621,394,956,199,689,075,738,676,622,921,702,928,106,039,370,003 = isqrt( 7^159 )
107,248,745,561,741,667,419,349,764,693,397,823,530,170,736,360,451,920,496,742,275,590,023 = isqrt( 7^161 )
750,741,218,932,191,671,935,448,352,853,784,764,711,195,154,523,163,443,477,195,929,130,162 = isqrt( 7^163 )
5,255,188,532,525,341,703,548,138,469,976,493,352,978,366,081,662,144,104,340,371,503,911,136 = isqrt( 7^165 )
36,786,319,727,677,391,924,836,969,289,835,453,470,848,562,571,635,008,730,382,600,527,377,954 = isqrt( 7^167 )
257,504,238,093,741,743,473,858,785,028,848,174,295,939,938,001,445,061,112,678,203,691,645,679 = isqrt( 7^169 )
1,802,529,666,656,192,204,317,011,495,201,937,220,071,579,566,010,115,427,788,747,425,841,519,758 = isqrt( 7^171 )
12,617,707,666,593,345,430,219,080,466,413,560,540,501,056,962,070,807,994,521,231,980,890,638,309 = isqrt( 7^173 )
88,323,953,666,153,418,011,533,563,264,894,923,783,507,398,734,495,655,961,648,623,866,234,468,168 = isqrt( 7^175 )
618,267,675,663,073,926,080,734,942,854,264,466,484,551,791,141,469,591,731,540,367,063,641,277,182 = isqrt( 7^177 )
4,327,873,729,641,517,482,565,144,599,979,851,265,391,862,537,990,287,142,120,782,569,445,488,940,274 = isqrt( 7^179 )
30,295,116,107,490,622,377,956,012,199,858,958,857,743,037,765,932,009,994,845,477,986,118,422,581,921 = isqrt( 7^181 )
212,065,812,752,434,356,645,692,085,399,012,712,004,201,264,361,524,069,963,918,345,902,828,958,073,452 = isqrt( 7^183 )
1,484,460,689,267,040,496,519,844,597,793,088,984,029,408,850,530,668,489,747,428,421,319,802,706,514,166 = isqrt( 7^185 )
10,391,224,824,869,283,475,638,912,184,551,622,888,205,861,953,714,679,428,231,998,949,238,618,945,599,162 = isqrt( 7^187 )
72,738,573,774,084,984,329,472,385,291,861,360,217,441,033,676,002,755,997,623,992,644,670,332,619,194,135 = isqrt( 7^189 )
509,170,016,418,594,890,306,306,697,043,029,521,522,087,235,732,019,291,983,367,948,512,692,328,334,358,945 = isqrt( 7^191 )
3,564,190,114,930,164,232,144,146,879,301,206,650,654,610,650,124,135,043,883,575,639,588,846,298,340,512,620 = isqrt( 7^193 )
24,949,330,804,511,149,625,009,028,155,108,446,554,582,274,550,868,945,307,185,029,477,121,924,088,383,588,341 = isqrt( 7^195 )
174,645,315,631,578,047,375,063,197,085,759,125,882,075,921,856,082,617,150,295,206,339,853,468,618,685,118,393 = isqrt( 7^197 )
1,222,517,209,421,046,331,625,442,379,600,313,881,174,531,452,992,578,320,052,066,444,378,974,280,330,795,828,756 = isqrt( 7^199 )
8,557,620,465,947,324,321,378,096,657,202,197,168,221,720,170,948,048,240,364,465,110,652,819,962,315,570,801,294 = isqrt( 7^201 )
59,903,343,261,631,270,249,646,676,600,415,380,177,552,041,196,636,337,682,551,255,774,569,739,736,208,995,609,059 = isqrt( 7^203 )
Quackery[edit]
[ dup size 3 / times
[ char , swap
i 1+ -3 * stuff ]
dup 0 peek char , =
if [ behead drop ] ] is +commas ( $ --> $ )
[ over size -
space swap of
swap join ] is justify ( $ n --> $ )
[ 1
[ 2dup < not while
2 << again ]
0
[ over 1 > while
dip
[ 2 >>
2dup - ]
dup 1 >>
unrot -
dup 0 < iff drop
else
[ 2swap nip
rot over + ]
again ]
nip swap ] is sqrt ( n --> n n )
( sqrt returns the integer square root and remainder )
( i.e. isqrt of 28 is 5 remainder 3 as (5^2)+3 = 28 )
( To make it task compliant change the last line to )
( "nip nip ] is sqrt ( n --> n )" )
66 times [ i^ sqrt drop echo sp ] cr cr
73 times
[ 7 i^ 1+ ** sqrt drop
number$ +commas 41 justify
echo$ cr
2 step ]
Output:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8
2
18
129
907
6,352
44,467
311,269
2,178,889
15,252,229
106,765,608
747,359,260
5,231,514,822
36,620,603,758
256,344,226,312
1,794,409,584,184
12,560,867,089,291
87,926,069,625,040
615,482,487,375,282
4,308,377,411,626,977
30,158,641,881,388,842
211,110,493,169,721,897
1,477,773,452,188,053,281
10,344,414,165,316,372,973
72,410,899,157,214,610,812
506,876,294,100,502,275,687
3,548,134,058,703,515,929,815
24,836,938,410,924,611,508,707
173,858,568,876,472,280,560,953
1,217,009,982,135,305,963,926,677
8,519,069,874,947,141,747,486,745
59,633,489,124,629,992,232,407,216
417,434,423,872,409,945,626,850,517
2,922,040,967,106,869,619,387,953,625
20,454,286,769,748,087,335,715,675,381
143,180,007,388,236,611,350,009,727,669
1,002,260,051,717,656,279,450,068,093,686
7,015,820,362,023,593,956,150,476,655,802
Racket[edit]
#lang racket
;; Integer Square Root (using Quadratic Residue)
(define (isqrt x)
(define q-init ; power of 4 greater than x
(let loop ([acc 1])
(if (<= acc x) (loop (* acc 4)) acc)))
(define-values (z r q)
(let loop ([z x] [r 0] [q q-init])
(if (<= q 1)
(values z r q)
(let* ([q (/ q 4)]
[t (- z r q)]
[r (/ r 2)])
(if (>= t 0)
(loop t (+ r q) q)
(loop z r q))))))
r)
(define (format-with-commas str #:chunk-size [size 3])
(define len (string-length str))
(define len-mod (modulo len size))
(define chunks
(for/list ([i (in-range len-mod len size)])
(substring str i (+ i size))))
(string-join (if (= len-mod 0)
chunks
(cons (substring str 0 len-mod) chunks))
","))
(displayln "Isqrt of integers (0 -> 65):")
(for ([i 66])
(printf "~a " (isqrt i)))
(displayln "\n\nIsqrt of odd powers of 7 (7 -> 7^73):")
(for/fold ([num 7]) ([i (in-range 1 74 2)])
(printf "Isqrt(7^~a) = ~a\n"
i
(format-with-commas (number->string (isqrt num))))
(* num 49))
- Output:
Isqrt of integers (0 -> 65): 0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 Isqrt of odd powers of 7 (7 -> 7^73): Isqrt(7^1) = 2 Isqrt(7^3) = 18 Isqrt(7^5) = 129 Isqrt(7^7) = 907 Isqrt(7^9) = 6,352 Isqrt(7^11) = 44,467 Isqrt(7^13) = 311,269 Isqrt(7^15) = 2,178,889 Isqrt(7^17) = 15,252,229 Isqrt(7^19) = 106,765,608 Isqrt(7^21) = 747,359,260 Isqrt(7^23) = 5,231,514,822 Isqrt(7^25) = 36,620,603,758 Isqrt(7^27) = 256,344,226,312 Isqrt(7^29) = 1,794,409,584,184 Isqrt(7^31) = 12,560,867,089,291 Isqrt(7^33) = 87,926,069,625,040 Isqrt(7^35) = 615,482,487,375,282 Isqrt(7^37) = 4,308,377,411,626,977 Isqrt(7^39) = 30,158,641,881,388,842 Isqrt(7^41) = 211,110,493,169,721,897 Isqrt(7^43) = 1,477,773,452,188,053,281 Isqrt(7^45) = 10,344,414,165,316,372,973 Isqrt(7^47) = 72,410,899,157,214,610,812 Isqrt(7^49) = 506,876,294,100,502,275,687 Isqrt(7^51) = 3,548,134,058,703,515,929,815 Isqrt(7^53) = 24,836,938,410,924,611,508,707 Isqrt(7^55) = 173,858,568,876,472,280,560,953 Isqrt(7^57) = 1,217,009,982,135,305,963,926,677 Isqrt(7^59) = 8,519,069,874,947,141,747,486,745 Isqrt(7^61) = 59,633,489,124,629,992,232,407,216 Isqrt(7^63) = 417,434,423,872,409,945,626,850,517 Isqrt(7^65) = 2,922,040,967,106,869,619,387,953,625 Isqrt(7^67) = 20,454,286,769,748,087,335,715,675,381 Isqrt(7^69) = 143,180,007,388,236,611,350,009,727,669 Isqrt(7^71) = 1,002,260,051,717,656,279,450,068,093,686 Isqrt(7^73) = 7,015,820,362,023,593,956,150,476,655,802
Raku[edit]
There is a task Integer roots that covers a similar operation, with the caveat that it will calculate any nth root (including 2) not just square roots.
See the Integer roots Raku entry.
Quadratic residue algorithm follows:
use Lingua::EN::Numbers;
sub isqrt ( \x ) { my ( $X, $q, $r, $t ) = x, 1, 0 ;
$q +<= 2 while $q ≤ $X ;
while $q > 1 {
$q +>= 2; $t = $X - $r - $q; $r +>= 1;
if $t ≥ 0 { $X = $t; $r += $q }
}
$r
}
say (^66)».&{ isqrt $_ }.Str ;
(1, 3…73)».&{ "7**$_: " ~ comma(isqrt 7**$_) }».say
- Output:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 7**1: 2 7**3: 18 7**5: 129 7**7: 907 7**9: 6,352 7**11: 44,467 7**13: 311,269 7**15: 2,178,889 7**17: 15,252,229 7**19: 106,765,608 7**21: 747,359,260 7**23: 5,231,514,822 7**25: 36,620,603,758 7**27: 256,344,226,312 7**29: 1,794,409,584,184 7**31: 12,560,867,089,291 7**33: 87,926,069,625,040 7**35: 615,482,487,375,282 7**37: 4,308,377,411,626,977 7**39: 30,158,641,881,388,842 7**41: 211,110,493,169,721,897 7**43: 1,477,773,452,188,053,281 7**45: 10,344,414,165,316,372,973 7**47: 72,410,899,157,214,610,812 7**49: 506,876,294,100,502,275,687 7**51: 3,548,134,058,703,515,929,815 7**53: 24,836,938,410,924,611,508,707 7**55: 173,858,568,876,472,280,560,953 7**57: 1,217,009,982,135,305,963,926,677 7**59: 8,519,069,874,947,141,747,486,745 7**61: 59,633,489,124,629,992,232,407,216 7**63: 417,434,423,872,409,945,626,850,517 7**65: 2,922,040,967,106,869,619,387,953,625 7**67: 20,454,286,769,748,087,335,715,675,381 7**69: 143,180,007,388,236,611,350,009,727,669 7**71: 1,002,260,051,717,656,279,450,068,093,686 7**73: 7,015,820,362,023,593,956,150,476,655,802
REXX[edit]
A fair amount of code was included so that the output aligns correctly.
/*REXX program computes and displays the Isqrt (integer square root) of some integers.*/
numeric digits 200 /*insure 'nuff decimal digs for results*/
parse arg range power base . /*obtain optional arguments from the CL*/
if range=='' | range=="," then range= 0..65 /*Not specified? Then use the default.*/
if power=='' | power=="," then power= 1..73 /* " " " " " " */
if base =='' | base =="," then base = 7 /* " " " " " " */
parse var range rLO '..' rHI; if rHI=='' then rHI= rLO /*handle a range? */
parse var power pLO '..' pHI; if pHI=='' then pHI= pLO /* " " " */
$=
do j=rLO to rHI while rHI>0 /*compute Isqrt for a range of integers*/
$= $ commas( Isqrt(j) ) /*append the Isqrt to a list for output*/
end /*j*/
$= strip($) /*elide the leading blank in the list. */
say center(' Isqrt for numbers: ' rLO " ──► " rHI' ', length($), "─")
say strip($) /*$ has a leading blank for 1st number*/
say
z= base ** pHI /*compute max. exponentiation product.*/
Lp= max(30, length( commas( z) ) ) /*length of " " " */
Lr= max(20, length( commas( Isqrt(z) ) ) ) /* " " " " " Isqrt of above.*/
say 'index' center(base"**index", Lp) center('Isqrt', Lr) /*show a title.*/
say '─────' copies("─", Lp) copies('─', Lr) /* " " header*/
do j=pLO to pHI by 2 while pHI>0; x= base ** j
say center(j, 5) right( commas(x), Lp) right( commas( Isqrt(x) ), Lr)
end /*j*/ /* [↑] show a bunch of powers & Isqrt.*/
exit 0 /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: parse arg _; do jc=length(_)-3 to 1 by -3; _=insert(',', _, jc); end; return _
/*──────────────────────────────────────────────────────────────────────────────────────*/
Isqrt: procedure; parse arg x /*obtain the only passed argument X. */
x= x % 1 /*convert possible real X to an integer*/ /* ◄■■■■■■■ optional. */
q= 1 /*initialize the Q variable to unity.*/
do until q>x /*find a Q that is greater than X. */
q= q * 4 /*multiply Q by four. */
end /*until*/
r= 0 /*R: will be the integer sqrt of X. */
do while q>1 /*keep processing while Q is > than 1*/
q= q % 4 /*divide Q by four (no remainder). */
t= x - r - q /*compute a temporary variable. */
r= r % 2 /*divide R by two (no remainder). */
if t >= 0 then do /*if T is non─negative ... */
x= t /*recompute the value of X */
r= r + q /* " " " " R */
end
end /*while*/
return r /*return the integer square root of X. */
- output when using the default inputs:
───────────────────────────────────────────────── Isqrt for numbers: 0 ──► 65 ────────────────────────────────────────────────── 0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 index 7**index Isqrt ───── ────────────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────── 1 7 2 3 343 18 5 16,807 129 7 823,543 907 9 40,353,607 6,352 11 1,977,326,743 44,467 13 96,889,010,407 311,269 15 4,747,561,509,943 2,178,889 17 232,630,513,987,207 15,252,229 19 11,398,895,185,373,143 106,765,608 21 558,545,864,083,284,007 747,359,260 23 27,368,747,340,080,916,343 5,231,514,822 25 1,341,068,619,663,964,900,807 36,620,603,758 27 65,712,362,363,534,280,139,543 256,344,226,312 29 3,219,905,755,813,179,726,837,607 1,794,409,584,184 31 157,775,382,034,845,806,615,042,743 12,560,867,089,291 33 7,730,993,719,707,444,524,137,094,407 87,926,069,625,040 35 378,818,692,265,664,781,682,717,625,943 615,482,487,375,282 37 18,562,115,921,017,574,302,453,163,671,207 4,308,377,411,626,977 39 909,543,680,129,861,140,820,205,019,889,143 30,158,641,881,388,842 41 44,567,640,326,363,195,900,190,045,974,568,007 211,110,493,169,721,897 43 2,183,814,375,991,796,599,109,312,252,753,832,343 1,477,773,452,188,053,281 45 107,006,904,423,598,033,356,356,300,384,937,784,807 10,344,414,165,316,372,973 47 5,243,338,316,756,303,634,461,458,718,861,951,455,543 72,410,899,157,214,610,812 49 256,923,577,521,058,878,088,611,477,224,235,621,321,607 506,876,294,100,502,275,687 51 12,589,255,298,531,885,026,341,962,383,987,545,444,758,743 3,548,134,058,703,515,929,815 53 616,873,509,628,062,366,290,756,156,815,389,726,793,178,407 24,836,938,410,924,611,508,707 55 30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943 173,858,568,876,472,280,560,953 57 1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207 1,217,009,982,135,305,963,926,677 59 72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143 8,519,069,874,947,141,747,486,745 61 3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007 59,633,489,124,629,992,232,407,216 63 174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343 417,434,423,872,409,945,626,850,517 65 8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807 2,922,040,967,106,869,619,387,953,625 67 418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543 20,454,286,769,748,087,335,715,675,381 69 20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607 143,180,007,388,236,611,350,009,727,669 71 1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 1,002,260,051,717,656,279,450,068,093,686 73 49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 7,015,820,362,023,593,956,150,476,655,802
Ruby[edit]
Ruby already has Integer.sqrt, which results in the integer square root of a positive integer. It can be re-implemented as follows:
module Commatize
refine Integer do
def commatize
self.to_s.gsub( /(\d)(?=\d{3}+(?:\.|$))(\d{3}\..*)?/, "\\1,\\2")
end
end
end
using Commatize
def isqrt(x)
q, r = 1, 0
while (q <= x) do q <<= 2 end
while (q > 1) do
q >>= 2; t = x-r-q; r >>= 1
if (t >= 0) then x, r = t, r+q end
end
r
end
puts (0..65).map{|n| isqrt(n) }.join(" ")
1.step(73, 2) do |n|
print "#{n}:\t"
puts isqrt(7**n).commatize
end
- Output:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 1: 2 3: 18 5: 129 7: 907 9: 6,352 11: 44,467 13: 311,269 15: 2,178,889 17: 15,252,229 19: 106,765,608 21: 747,359,260 23: 5,231,514,822 25: 36,620,603,758 27: 256,344,226,312 29: 1,794,409,584,184 31: 12,560,867,089,291 33: 87,926,069,625,040 35: 615,482,487,375,282 37: 4,308,377,411,626,977 39: 30,158,641,881,388,842 41: 211,110,493,169,721,897 43: 1,477,773,452,188,053,281 45: 10,344,414,165,316,372,973 47: 72,410,899,157,214,610,812 49: 506,876,294,100,502,275,687 51: 3,548,134,058,703,515,929,815 53: 24,836,938,410,924,611,508,707 55: 173,858,568,876,472,280,560,953 57: 1,217,009,982,135,305,963,926,677 59: 8,519,069,874,947,141,747,486,745 61: 59,633,489,124,629,992,232,407,216 63: 417,434,423,872,409,945,626,850,517 65: 2,922,040,967,106,869,619,387,953,625 67: 20,454,286,769,748,087,335,715,675,381 69: 143,180,007,388,236,611,350,009,727,669 71: 1,002,260,051,717,656,279,450,068,093,686 73: 7,015,820,362,023,593,956,150,476,655,802
Rust[edit]
use num::BigUint;
use num::CheckedSub;
use num_traits::{One, Zero};
fn isqrt(number: &BigUint) -> BigUint {
let mut q: BigUint = One::one();
while q <= *number {
q <<= &2;
}
let mut z = number.clone();
let mut result: BigUint = Zero::zero();
while q > One::one() {
q >>= &2;
let t = z.checked_sub(&result).and_then(|diff| diff.checked_sub(&q));
result >>= &1;
if let Some(t) = t {
z = t;
result += &q;
}
}
result
}
fn with_thousand_separator(s: &str) -> String {
let digits: Vec<_> = s.chars().rev().collect();
let chunks: Vec<_> = digits
.chunks(3)
.map(|chunk| chunk.iter().collect::<String>())
.collect();
chunks.join(",").chars().rev().collect::<String>()
}
fn main() {
println!("The integer square roots of integers from 0 to 65 are:");
(0_u32..=65).for_each(|n| print!("{} ", isqrt(&n.into())));
println!("\nThe integer square roots of odd powers of 7 from 7^1 up to 7^74 are:");
(1_u32..75).step_by(2).for_each(|exp| {
println!(
"7^{:>2}={:>83} ISQRT: {:>42} ",
exp,
with_thousand_separator(&BigUint::from(7_u8).pow(exp).to_string()),
with_thousand_separator(&isqrt(&BigUint::from(7_u8).pow(exp)).to_string())
)
});
}
- Output:
The integer square roots of integers from 0 to 65 are: 0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 The integer square roots of odd powers of 7 from 7^1 up to 7^74 are: 7^ 1= 7 ISQRT: 2 7^ 3= 343 ISQRT: 18 7^ 5= 16,807 ISQRT: 129 7^ 7= 823,543 ISQRT: 907 7^ 9= 40,353,607 ISQRT: 6,352 7^11= 1,977,326,743 ISQRT: 44,467 7^13= 96,889,010,407 ISQRT: 311,269 7^15= 4,747,561,509,943 ISQRT: 2,178,889 7^17= 232,630,513,987,207 ISQRT: 15,252,229 7^19= 11,398,895,185,373,143 ISQRT: 106,765,608 7^21= 558,545,864,083,284,007 ISQRT: 747,359,260 7^23= 27,368,747,340,080,916,343 ISQRT: 5,231,514,822 7^25= 1,341,068,619,663,964,900,807 ISQRT: 36,620,603,758 7^27= 65,712,362,363,534,280,139,543 ISQRT: 256,344,226,312 7^29= 3,219,905,755,813,179,726,837,607 ISQRT: 1,794,409,584,184 7^31= 157,775,382,034,845,806,615,042,743 ISQRT: 12,560,867,089,291 7^33= 7,730,993,719,707,444,524,137,094,407 ISQRT: 87,926,069,625,040 7^35= 378,818,692,265,664,781,682,717,625,943 ISQRT: 615,482,487,375,282 7^37= 18,562,115,921,017,574,302,453,163,671,207 ISQRT: 4,308,377,411,626,977 7^39= 909,543,680,129,861,140,820,205,019,889,143 ISQRT: 30,158,641,881,388,842 7^41= 44,567,640,326,363,195,900,190,045,974,568,007 ISQRT: 211,110,493,169,721,897 7^43= 2,183,814,375,991,796,599,109,312,252,753,832,343 ISQRT: 1,477,773,452,188,053,281 7^45= 107,006,904,423,598,033,356,356,300,384,937,784,807 ISQRT: 10,344,414,165,316,372,973 7^47= 5,243,338,316,756,303,634,461,458,718,861,951,455,543 ISQRT: 72,410,899,157,214,610,812 7^49= 256,923,577,521,058,878,088,611,477,224,235,621,321,607 ISQRT: 506,876,294,100,502,275,687 7^51= 12,589,255,298,531,885,026,341,962,383,987,545,444,758,743 ISQRT: 3,548,134,058,703,515,929,815 7^53= 616,873,509,628,062,366,290,756,156,815,389,726,793,178,407 ISQRT: 24,836,938,410,924,611,508,707 7^55= 30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943 ISQRT: 173,858,568,876,472,280,560,953 7^57= 1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207 ISQRT: 1,217,009,982,135,305,963,926,677 7^59= 72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143 ISQRT: 8,519,069,874,947,141,747,486,745 7^61= 3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007 ISQRT: 59,633,489,124,629,992,232,407,216 7^63= 174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343 ISQRT: 417,434,423,872,409,945,626,850,517 7^65= 8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807 ISQRT: 2,922,040,967,106,869,619,387,953,625 7^67= 418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543 ISQRT: 20,454,286,769,748,087,335,715,675,381 7^69= 20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607 ISQRT: 143,180,007,388,236,611,350,009,727,669 7^71= 1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 ISQRT: 1,002,260,051,717,656,279,450,068,093,686 7^73= 49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 ISQRT: 7,015,820,362,023,593,956,150,476,655,802
Seed7[edit]
$ include "seed7_05.s7i";
include "bigint.s7i";
const func string: commatize (in bigInteger: bigNum) is func
result
var string: stri is "";
local
var integer: index is 0;
begin
stri := str(bigNum);
for index range length(stri) - 3 downto 1 step 3 do
stri := stri[.. index] & "," & stri[succ(index) ..];
end for;
end func;
const proc: main is func
local
var integer: number is 0;
var bigInteger: pow7 is 7_;
begin
writeln("The integer square roots of integers from 0 to 65 are:");
for number range 0 to 65 do
write(sqrt(number) <& " ");
end for;
writeln("\n\nThe integer square roots of powers of 7 from 7**1 up to 7**73 are:");
writeln("power 7 ** power integer square root");
writeln("----- --------------------------------------------------------------------------------- -----------------------------------------");
for number range 1 to 73 step 2 do
writeln(number lpad 2 <& commatize(pow7) lpad 85 <& commatize(sqrt(pow7)) lpad 42);
pow7 := pow7 * 49_;
end for;
end func;
- Output:
The integer square roots of integers from 0 to 65 are: 0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 The integer square roots of powers of 7 from 7**1 up to 7**73 are: power 7 ** power integer square root ----- --------------------------------------------------------------------------------- ----------------------------------------- 1 7 2 3 343 18 5 16,807 129 7 823,543 907 9 40,353,607 6,352 11 1,977,326,743 44,467 13 96,889,010,407 311,269 15 4,747,561,509,943 2,178,889 17 232,630,513,987,207 15,252,229 19 11,398,895,185,373,143 106,765,608 21 558,545,864,083,284,007 747,359,260 23 27,368,747,340,080,916,343 5,231,514,822 25 1,341,068,619,663,964,900,807 36,620,603,758 27 65,712,362,363,534,280,139,543 256,344,226,312 29 3,219,905,755,813,179,726,837,607 1,794,409,584,184 31 157,775,382,034,845,806,615,042,743 12,560,867,089,291 33 7,730,993,719,707,444,524,137,094,407 87,926,069,625,040 35 378,818,692,265,664,781,682,717,625,943 615,482,487,375,282 37 18,562,115,921,017,574,302,453,163,671,207 4,308,377,411,626,977 39 909,543,680,129,861,140,820,205,019,889,143 30,158,641,881,388,842 41 44,567,640,326,363,195,900,190,045,974,568,007 211,110,493,169,721,897 43 2,183,814,375,991,796,599,109,312,252,753,832,343 1,477,773,452,188,053,281 45 107,006,904,423,598,033,356,356,300,384,937,784,807 10,344,414,165,316,372,973 47 5,243,338,316,756,303,634,461,458,718,861,951,455,543 72,410,899,157,214,610,812 49 256,923,577,521,058,878,088,611,477,224,235,621,321,607 506,876,294,100,502,275,687 51 12,589,255,298,531,885,026,341,962,383,987,545,444,758,743 3,548,134,058,703,515,929,815 53 616,873,509,628,062,366,290,756,156,815,389,726,793,178,407 24,836,938,410,924,611,508,707 55 30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943 173,858,568,876,472,280,560,953 57 1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207 1,217,009,982,135,305,963,926,677 59 72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143 8,519,069,874,947,141,747,486,745 61 3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007 59,633,489,124,629,992,232,407,216 63 174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343 417,434,423,872,409,945,626,850,517 65 8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807 2,922,040,967,106,869,619,387,953,625 67 418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543 20,454,286,769,748,087,335,715,675,381 69 20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607 143,180,007,388,236,611,350,009,727,669 71 1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 1,002,260,051,717,656,279,450,068,093,686 73 49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 7,015,820,362,023,593,956,150,476,655,802
Sidef[edit]
Built-in:
var n = 1234
say n.isqrt
say n.iroot(2)
Explicit implementation for the integer k-th root of n:
func rootint(n, k=2) {
return 0 if (n == 0)
var (s, v) = (n, k - 1)
loop {
var u = ((v*s + (n // s**v)) // k)
break if (u >= s)
s = u
}
s
}
Implementation of integer square root of n (using the quadratic residue algorithm):
func isqrt(x) { var (q, r) = (1, 0); while (q <= x) { q <<= 2 }
while (q > 1) { q >>= 2; var t = x-r+q; r >>= 1
if (t >= 0) { (x, r) = (t, r+q) } } r }
say isqrt.map(0..65).join(' '); printf("\n")
for n in (1..73 `by` 2) {
printf("isqrt(7^%-2d): %42s\n", n, isqrt(7**n).commify) }
- Output:
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 isqrt(7^1 ): 2 isqrt(7^3 ): 18 isqrt(7^5 ): 129 isqrt(7^7 ): 907 isqrt(7^9 ): 6,352 isqrt(7^11): 44,467 isqrt(7^13): 311,269 isqrt(7^15): 2,178,889 isqrt(7^17): 15,252,229 isqrt(7^19): 106,765,608 isqrt(7^21): 747,359,260 isqrt(7^23): 5,231,514,822 isqrt(7^25): 36,620,603,758 isqrt(7^27): 256,344,226,312 isqrt(7^29): 1,794,409,584,184 isqrt(7^31): 12,560,867,089,291 isqrt(7^33): 87,926,069,625,040 isqrt(7^35): 615,482,487,375,282 isqrt(7^37): 4,308,377,411,626,977 isqrt(7^39): 30,158,641,881,388,842 isqrt(7^41): 211,110,493,169,721,897 isqrt(7^43): 1,477,773,452,188,053,281 isqrt(7^45): 10,344,414,165,316,372,973 isqrt(7^47): 72,410,899,157,214,610,812 isqrt(7^49): 506,876,294,100,502,275,687 isqrt(7^51): 3,548,134,058,703,515,929,815 isqrt(7^53): 24,836,938,410,924,611,508,707 isqrt(7^55): 173,858,568,876,472,280,560,953 isqrt(7^57): 1,217,009,982,135,305,963,926,677 isqrt(7^59): 8,519,069,874,947,141,747,486,745 isqrt(7^61): 59,633,489,124,629,992,232,407,216 isqrt(7^63): 417,434,423,872,409,945,626,850,517 isqrt(7^65): 2,922,040,967,106,869,619,387,953,625 isqrt(7^67): 20,454,286,769,748,087,335,715,675,381 isqrt(7^69): 143,180,007,388,236,611,350,009,727,669 isqrt(7^71): 1,002,260,051,717,656,279,450,068,093,686 isqrt(7^73): 7,015,820,362,023,593,956,150,476,655,802
Swift[edit]
Requires the attaswift BigInt package.
import BigInt
func integerSquareRoot<T: BinaryInteger>(_ num: T) -> T {
var x: T = num
var q: T = 1
while q <= x {
q <<= 2
}
var r: T = 0
while q > 1 {
q >>= 2
let t: T = x - r - q
r >>= 1
if t >= 0 {
x = t
r += q
}
}
return r
}
func pad(string: String, width: Int) -> String {
if string.count >= width {
return string
}
return String(repeating: " ", count: width - string.count) + string
}
func commatize<T: BinaryInteger>(_ num: T) -> String {
let string = String(num)
var result = String()
result.reserveCapacity(4 * string.count / 3)
var i = 0
for ch in string {
if i > 0 && i % 3 == string.count % 3 {
result += ","
}
result.append(ch)
i += 1
}
return result
}
print("Integer square root for numbers 0 to 65:")
for n in 0...65 {
print(integerSquareRoot(n), terminator: " ")
}
let powerWidth = 83
let isqrtWidth = 42
print("\n\nInteger square roots of odd powers of 7 from 1 to 73:")
print(" n |\(pad(string: "7 ^ n", width: powerWidth)) |\(pad(string: "isqrt(7 ^ n)", width: isqrtWidth))")
print(String(repeating: "-", count: powerWidth + isqrtWidth + 6))
var p: BigInt = 7
for n in stride(from: 1, through: 73, by: 2) {
let power = pad(string: commatize(p), width: powerWidth)
let isqrt = pad(string: commatize(integerSquareRoot(p)), width: isqrtWidth)
print("\(pad(string: String(n), width: 2)) |\(power) |\(isqrt)")
p *= 49
}
- Output:
Integer square root for numbers 0 to 65: 0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 Integer square roots of odd powers of 7 from 1 to 73: n | 7 ^ n | isqrt(7 ^ n) ----------------------------------------------------------------------------------------------------------------------------------- 1 | 7 | 2 3 | 343 | 18 5 | 16,807 | 129 7 | 823,543 | 907 9 | 40,353,607 | 6,352 11 | 1,977,326,743 | 44,467 13 | 96,889,010,407 | 311,269 15 | 4,747,561,509,943 | 2,178,889 17 | 232,630,513,987,207 | 15,252,229 19 | 11,398,895,185,373,143 | 106,765,608 21 | 558,545,864,083,284,007 | 747,359,260 23 | 27,368,747,340,080,916,343 | 5,231,514,822 25 | 1,341,068,619,663,964,900,807 | 36,620,603,758 27 | 65,712,362,363,534,280,139,543 | 256,344,226,312 29 | 3,219,905,755,813,179,726,837,607 | 1,794,409,584,184 31 | 157,775,382,034,845,806,615,042,743 | 12,560,867,089,291 33 | 7,730,993,719,707,444,524,137,094,407 | 87,926,069,625,040 35 | 378,818,692,265,664,781,682,717,625,943 | 615,482,487,375,282 37 | 18,562,115,921,017,574,302,453,163,671,207 | 4,308,377,411,626,977 39 | 909,543,680,129,861,140,820,205,019,889,143 | 30,158,641,881,388,842 41 | 44,567,640,326,363,195,900,190,045,974,568,007 | 211,110,493,169,721,897 43 | 2,183,814,375,991,796,599,109,312,252,753,832,343 | 1,477,773,452,188,053,281 45 | 107,006,904,423,598,033,356,356,300,384,937,784,807 | 10,344,414,165,316,372,973 47 | 5,243,338,316,756,303,634,461,458,718,861,951,455,543 | 72,410,899,157,214,610,812 49 | 256,923,577,521,058,878,088,611,477,224,235,621,321,607 | 506,876,294,100,502,275,687 51 | 12,589,255,298,531,885,026,341,962,383,987,545,444,758,743 | 3,548,134,058,703,515,929,815 53 | 616,873,509,628,062,366,290,756,156,815,389,726,793,178,407 | 24,836,938,410,924,611,508,707 55 | 30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943 | 173,858,568,876,472,280,560,953 57 | 1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207 | 1,217,009,982,135,305,963,926,677 59 | 72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143 | 8,519,069,874,947,141,747,486,745 61 | 3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007 | 59,633,489,124,629,992,232,407,216 63 | 174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343 | 417,434,423,872,409,945,626,850,517 65 | 8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807 | 2,922,040,967,106,869,619,387,953,625 67 | 418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543 | 20,454,286,769,748,087,335,715,675,381 69 | 20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607 | 143,180,007,388,236,611,350,009,727,669 71 | 1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 | 1,002,260,051,717,656,279,450,068,093,686 73 | 49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 | 7,015,820,362,023,593,956,150,476,655,802
Tiny BASIC[edit]
Tiny BASIC does not support string formatting or concatenation, and is limited to integer arithmetic on numbers no greater than 32,767. The isqrt of 0-65 and the first two odd powers of 7 are shown in column format. The algorithm itself (the interesting part) begins on line 100.
10 LET X = 0
20 GOSUB 100
30 PRINT R
40 LET X = X + 1
50 IF X < 66 THEN GOTO 20
70 PRINT "---"
71 LET X = 7
72 GOSUB 100
73 PRINT R
77 LET X = 343
78 GOSUB 100
79 PRINT R
90 END
100 REM integer square root function
110 LET Q = 1
120 IF Q > X THEN GOTO 150
130 LET Q = Q * 4
140 GOTO 120
150 LET Z = X
160 LET R = 0
170 IF Q <= 1 THEN RETURN
180 LET Q = Q / 4
190 LET T = Z - R - Q
200 LET R = R / 2
210 IF T < 0 THEN GOTO 170
220 LET Z = T
230 LET R = R + Q
240 GOTO 170
UNIX Shell[edit]
function isqrt {
typeset -i x
for x; do
typeset -i q=1
while (( q <= x )); do
(( q <<= 2 ))
if (( q <= 0 )); then
return 1
fi
done
typeset -i z=x
typeset -i r=0
typeset -i t
while (( q > 1 )); do
(( q >>= 2 ))
(( t = z - r - q ))
(( r >>= 1 ))
if (( t >= 0 )); then
(( z = t ))
(( r = r + q ))
fi
done
printf '%d\n' "$r"
done
}
# demo
printf 'isqrt(n) for n from 0 to 65:\n'
for i in {1..4}; do
for n in {0..65}; do
case $i in
1)
(( tens=n/10 ))
if (( tens )); then
printf '%2d' "$tens"
else
printf ' '
fi
;;
2) printf '%2d' $(( n%10 ));;
3) printf -- '--';;
4) printf '%2d' "$(isqrt "$n")";;
esac
done
printf '\n'
done
printf '\n'
printf 'isqrt(7ⁿ) for odd n up to the limit of integer precision:\n'
printf '%2s|%27sⁿ|%14sⁿ)\n' "n" "7" "isqrt(7"
for (( i=0;i<48; ++i )); do printf '-'; done; printf '\n'
for (( p=1; p<=73 && (n=7**p) > 0; p+=2)); do
if r=$(isqrt $n); then
printf "%2d|%'28d|%'16d\n" "$p" "$n" "$r"
else
break
fi
done
- Output:
The powers-of-7 table is limited by the built-in precision; on my system, both bash and zsh use signed 64-bit integers with a max value of 7²² < 9223372036854775807 < 7²³. Ksh uses signed 32-bit integers with a max value of 7¹¹ < 2147483647 < 7¹²; if I remove the typeset -i integer restriction, the code will work to a much larger power of 7, but at that point it's doing floating-point arithmetic, which is against the spirit of the task.
isqrt(n) for n from 0 to 65:
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
------------------------------------------------------------------------------------------------------------------------------------
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8
isqrt(7ⁿ) for odd n up to the limit of integer precision:
n| 7ⁿ| isqrt(7ⁿ)
------------------------------------------------
1| 7| 2
3| 343| 18
5| 16,807| 129
7| 823,543| 907
9| 40,353,607| 6,352 # ksh stops here
11| 1,977,326,743| 44,467
13| 96,889,010,407| 311,269
15| 4,747,561,509,943| 2,178,889
17| 232,630,513,987,207| 15,252,229
19| 11,398,895,185,373,143| 106,765,608
21| 558,545,864,083,284,007| 747,359,260
Visual Basic .NET[edit]
Imports System
Imports System.Console
Imports BI = System.Numerics.BigInteger
Module Module1
Function isqrt(ByVal x As BI) As BI
Dim t As BI, q As BI = 1, r As BI = 0
While q <= x : q <<= 2 : End While
While q > 1 : q >>= 2 : t = x - r - q : r >>= 1
If t >= 0 Then x = t : r += q
End While : Return r
End Function
Sub Main()
Const max As Integer = 73, smax As Integer = 65
Dim power_width As Integer = ((BI.Pow(7, max).ToString().Length \ 3) << 2) + 3,
isqrt_width As Integer = (power_width + 1) >> 1,
n as Integer
WriteLine("Integer square root for numbers 0 to {0}:", smax)
For n = 0 To smax : Write("{0} ", (n \ 10).ToString().Replace("0", " "))
Next : WriteLine()
For n = 0 To smax : Write("{0} ", n Mod 10) : Next : WriteLine()
WriteLine(New String("-"c, (smax << 1) + 1))
For n = 0 To smax : Write("{0} ", isqrt(n)) : Next
WriteLine(vbLf & vbLf & "Integer square roots of odd powers of 7 from 1 to {0}:", max)
Dim s As String = String.Format("[0,2] |[1,{0}:n0] |[2,{1}:n0]",
power_width, isqrt_width).Replace("[", "{").Replace("]", "}")
WriteLine(s, "n", "7 ^ n", "isqrt(7 ^ n)")
WriteLine(New String("-"c, power_width + isqrt_width + 6))
Dim p As BI = 7 : n = 1 : While n <= max
WriteLine(s, n, p, isqrt(p)) : n += 2 : p = p * 49
End While
End Sub
End Module
- Output:
Integer square root for numbers 0 to 65:
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
-----------------------------------------------------------------------------------------------------------------------------------
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8
Integer square roots of odd powers of 7 from 1 to 73:
n | 7 ^ n | isqrt(7 ^ n)
-----------------------------------------------------------------------------------------------------------------------------------
1 | 7 | 2
3 | 343 | 18
5 | 16,807 | 129
7 | 823,543 | 907
9 | 40,353,607 | 6,352
11 | 1,977,326,743 | 44,467
13 | 96,889,010,407 | 311,269
15 | 4,747,561,509,943 | 2,178,889
17 | 232,630,513,987,207 | 15,252,229
19 | 11,398,895,185,373,143 | 106,765,608
21 | 558,545,864,083,284,007 | 747,359,260
23 | 27,368,747,340,080,916,343 | 5,231,514,822
25 | 1,341,068,619,663,964,900,807 | 36,620,603,758
27 | 65,712,362,363,534,280,139,543 | 256,344,226,312
29 | 3,219,905,755,813,179,726,837,607 | 1,794,409,584,184
31 | 157,775,382,034,845,806,615,042,743 | 12,560,867,089,291
33 | 7,730,993,719,707,444,524,137,094,407 | 87,926,069,625,040
35 | 378,818,692,265,664,781,682,717,625,943 | 615,482,487,375,282
37 | 18,562,115,921,017,574,302,453,163,671,207 | 4,308,377,411,626,977
39 | 909,543,680,129,861,140,820,205,019,889,143 | 30,158,641,881,388,842
41 | 44,567,640,326,363,195,900,190,045,974,568,007 | 211,110,493,169,721,897
43 | 2,183,814,375,991,796,599,109,312,252,753,832,343 | 1,477,773,452,188,053,281
45 | 107,006,904,423,598,033,356,356,300,384,937,784,807 | 10,344,414,165,316,372,973
47 | 5,243,338,316,756,303,634,461,458,718,861,951,455,543 | 72,410,899,157,214,610,812
49 | 256,923,577,521,058,878,088,611,477,224,235,621,321,607 | 506,876,294,100,502,275,687
51 | 12,589,255,298,531,885,026,341,962,383,987,545,444,758,743 | 3,548,134,058,703,515,929,815
53 | 616,873,509,628,062,366,290,756,156,815,389,726,793,178,407 | 24,836,938,410,924,611,508,707
55 | 30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943 | 173,858,568,876,472,280,560,953
57 | 1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207 | 1,217,009,982,135,305,963,926,677
59 | 72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143 | 8,519,069,874,947,141,747,486,745
61 | 3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007 | 59,633,489,124,629,992,232,407,216
63 | 174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343 | 417,434,423,872,409,945,626,850,517
65 | 8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807 | 2,922,040,967,106,869,619,387,953,625
67 | 418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543 | 20,454,286,769,748,087,335,715,675,381
69 | 20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607 | 143,180,007,388,236,611,350,009,727,669
71 | 1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 | 1,002,260,051,717,656,279,450,068,093,686
73 | 49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 | 7,015,820,362,023,593,956,150,476,655,802
Wren[edit]
import "/big" for BigInt
import "/fmt" for Fmt
var isqrt = Fn.new { |x|
if (!(x is BigInt && x >= BigInt.zero)) {
Fiber.abort("Argument must be a non-negative big integer.")
}
var q = BigInt.one
while (q <= x) q = q * 4
var z = x
var r = BigInt.zero
while (q > BigInt.one) {
q = q >> 2
var t = z - r - q
r = r >> 1
if (t >= 0) {
z = t
r = r + q
}
}
return r
}
System.print("The integer square roots of integers from 0 to 65 are:")
for (i in 0..65) System.write("%(isqrt.call(BigInt.new(i))) ")
System.print()
System.print("\nThe integer square roots of powers of 7 from 7^1 up to 7^73 are:\n")
System.print("power 7 ^ power integer square root")
System.print("----- --------------------------------------------------------------------------------- -----------------------------------------")
var pow7 = BigInt.new(7)
var bi49 = BigInt.new(49)
var i = 1
while (i <= 73) {
Fmt.print("$2d $,84s $,41s", i, pow7, isqrt.call(pow7))
pow7 = pow7 * bi49
i = i + 2
}
- Output:
The integer square roots of integers from 0 to 65 are: 0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 The integer square roots of odd powers of 7 from 7^1 up to 7^73 are: power 7 ^ power integer square root ----- --------------------------------------------------------------------------------- ----------------------------------------- 1 7 2 3 343 18 5 16,807 129 7 823,543 907 9 40,353,607 6,352 11 1,977,326,743 44,467 13 96,889,010,407 311,269 15 4,747,561,509,943 2,178,889 17 232,630,513,987,207 15,252,229 19 11,398,895,185,373,143 106,765,608 21 558,545,864,083,284,007 747,359,260 23 27,368,747,340,080,916,343 5,231,514,822 25 1,341,068,619,663,964,900,807 36,620,603,758 27 65,712,362,363,534,280,139,543 256,344,226,312 29 3,219,905,755,813,179,726,837,607 1,794,409,584,184 31 157,775,382,034,845,806,615,042,743 12,560,867,089,291 33 7,730,993,719,707,444,524,137,094,407 87,926,069,625,040 35 378,818,692,265,664,781,682,717,625,943 615,482,487,375,282 37 18,562,115,921,017,574,302,453,163,671,207 4,308,377,411,626,977 39 909,543,680,129,861,140,820,205,019,889,143 30,158,641,881,388,842 41 44,567,640,326,363,195,900,190,045,974,568,007 211,110,493,169,721,897 43 2,183,814,375,991,796,599,109,312,252,753,832,343 1,477,773,452,188,053,281 45 107,006,904,423,598,033,356,356,300,384,937,784,807 10,344,414,165,316,372,973 47 5,243,338,316,756,303,634,461,458,718,861,951,455,543 72,410,899,157,214,610,812 49 256,923,577,521,058,878,088,611,477,224,235,621,321,607 506,876,294,100,502,275,687 51 12,589,255,298,531,885,026,341,962,383,987,545,444,758,743 3,548,134,058,703,515,929,815 53 616,873,509,628,062,366,290,756,156,815,389,726,793,178,407 24,836,938,410,924,611,508,707 55 30,226,801,971,775,055,948,247,051,683,954,096,612,865,741,943 173,858,568,876,472,280,560,953 57 1,481,113,296,616,977,741,464,105,532,513,750,734,030,421,355,207 1,217,009,982,135,305,963,926,677 59 72,574,551,534,231,909,331,741,171,093,173,785,967,490,646,405,143 8,519,069,874,947,141,747,486,745 61 3,556,153,025,177,363,557,255,317,383,565,515,512,407,041,673,852,007 59,633,489,124,629,992,232,407,216 63 174,251,498,233,690,814,305,510,551,794,710,260,107,945,042,018,748,343 417,434,423,872,409,945,626,850,517 65 8,538,323,413,450,849,900,970,017,037,940,802,745,289,307,058,918,668,807 2,922,040,967,106,869,619,387,953,625 67 418,377,847,259,091,645,147,530,834,859,099,334,519,176,045,887,014,771,543 20,454,286,769,748,087,335,715,675,381 69 20,500,514,515,695,490,612,229,010,908,095,867,391,439,626,248,463,723,805,607 143,180,007,388,236,611,350,009,727,669 71 1,004,525,211,269,079,039,999,221,534,496,697,502,180,541,686,174,722,466,474,743 1,002,260,051,717,656,279,450,068,093,686 73 49,221,735,352,184,872,959,961,855,190,338,177,606,846,542,622,561,400,857,262,407 7,015,820,362,023,593,956,150,476,655,802
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