Quadrat special primes
- Task
n is smallest prime such that the difference of successive terms are the smallest squares of positive integers,
where n < 16000.
ALGOL 68
<lang algol68>BEGIN # find some primes where the gap between the current prime and the next is a square #
# reurns a sieve of primes up to n #
PROC eratosthenes = ( INT n )[]BOOL:
BEGIN
[ 1 : n ]BOOL p;
p[ 1 ] := FALSE; p[ 2 ] := TRUE;
FOR i FROM 3 BY 2 TO n DO p[ i ] := TRUE OD;
FOR i FROM 4 BY 2 TO n DO p[ i ] := FALSE OD;
FOR i FROM 3 BY 2 TO ENTIER sqrt( n ) DO
IF p[ i ] THEN FOR s FROM i * i BY i + i TO n DO p[ s ] := FALSE OD FI
OD;
p
END # eratosthenes # ;
# an array of squares #
PROC get squares = ( INT n )[]INT:
BEGIN
[ 1 : n ]INT s;
FOR i TO n DO s[ i ] := i * i OD;
s
END # get squares # ;
INT max number = 16000;
[]BOOL prime = eratosthenes( max number );
[]INT square = get squares( max number );
INT p count, this prime, next prime;
# the first square gap is 1 (between 2 and 3) the gap between all other primes is even #
# so we treat 2-3 as a special case #
p count := 1; this prime := 2; next prime := 3;
print( ( " ", whole( this prime, -5 ) ) );
WHILE next prime < max number DO
this prime := next prime;
p count +:= 1;
print( ( " ", whole( this prime, -5 ) ) );
IF p count MOD 12 = 0 THEN print( ( newline ) ) FI;
INT sq pos := 2;
WHILE next prime := this prime + square[ sq pos ];
IF next prime < max number THEN NOT prime[ next prime ] ELSE FALSE FI
DO sq pos +:= 2 OD
OD
END</lang>
- Output:
2 3 7 11 47 83 227 263 587 911 947 983 1019 1163 1307 1451 1487 1523 1559 2459 3359 4259 4583 5483 5519 5843 5879 6203 6779 7103 7247 7283 7607 7643 8219 8363 10667 11243 11279 11423 12323 12647 12791 13367 13691 14591 14627 14771 15671
ALGOL W
<lang algolw>begin % find some primes where the gap between the current prime and the next is a square %
% sets p( 1 :: n ) to a sieve of primes up to n %
procedure Eratosthenes ( logical array p( * ) ; integer value n ) ;
begin
p( 1 ) := false; p( 2 ) := true;
for i := 3 step 2 until n do p( i ) := true;
for i := 4 step 2 until n do p( i ) := false;
for i := 3 step 2 until truncate( sqrt( n ) ) do begin
integer ii; ii := i + i;
if p( i ) then for pr := i * i step ii until n do p( pr ) := false
end for_i ;
end Eratosthenes ;
% sets s( 1 :: n ) to the squares %
procedure getSquares ( integer array s ( * ) ; integer value n ) ;
for i := 1 until n do s( i ) := i * i;
integer MAX_NUMBER;
MAX_NUMBER := 16000;
begin
logical array prime( 1 :: MAX_NUMBER );
integer array square( 1 :: MAX_NUMBER );
integer pCount, thisPrime, nextPrime;
% sieve the primes to MAX_NUMBER %
Eratosthenes( prime, MAX_NUMBER );
% calculate the squares to MAX_NUMBER %
getSquares( square, MAX_NUMBER );
% the first gap is 1 (between 2 and 3) the gap between all other primes is even %
% so we treat 2-3 as a special case %
pCount := 1; thisPrime := 2; nextPrime := 3;
write( i_w := 6, s_w := 0, " ", thisPrime );
while nextPrime < MAX_NUMBER do begin
integer sqPos;
thisPrime := nextPrime;
pCount := pCount + 1;
writeon( i_w := 6, s_w := 0, " ", thisPrime );
if pCount rem 12 = 0 then write();
sqPos := 2;
while begin
nextPrime := thisPrime + square( sqPos );
nextPrime < MAX_NUMBER and not prime( nextPrime )
end do sqPos := sqPos + 2;
end while_thisPrime_lt_MAX_NUMBER
end
end.</lang>
- Output:
2 3 7 11 47 83 227 263 587 911 947 983 1019 1163 1307 1451 1487 1523 1559 2459 3359 4259 4583 5483 5519 5843 5879 6203 6779 7103 7247 7283 7607 7643 8219 8363 10667 11243 11279 11423 12323 12647 12791 13367 13691 14591 14627 14771 15671
Go
<lang go>package main
import (
"fmt" "math"
)
func sieve(limit int) []bool {
limit++
// True denotes composite, false denotes prime.
c := make([]bool, limit) // all false by default
c[0] = true
c[1] = true
// no need to bother with even numbers over 2 for this task
p := 3 // Start from 3.
for {
p2 := p * p
if p2 >= limit {
break
}
for i := p2; i < limit; i += 2 * p {
c[i] = true
}
for {
p += 2
if !c[p] {
break
}
}
}
return c
}
func isSquare(n int) bool {
s := int(math.Sqrt(float64(n))) return s*s == n
}
func commas(n int) string {
s := fmt.Sprintf("%d", n)
if n < 0 {
s = s[1:]
}
le := len(s)
for i := le - 3; i >= 1; i -= 3 {
s = s[0:i] + "," + s[i:]
}
if n >= 0 {
return s
}
return "-" + s
}
func main() {
c := sieve(15999)
fmt.Println("Quadrat special primes under 16,000:")
fmt.Println(" Prime1 Prime2 Gap Sqrt")
lastQuadSpecial := 3
gap := 1
count := 1
fmt.Printf("%7d %7d %6d %4d\n", 2, 3, 1, 1)
for i := 5; i < 16000; i += 2 {
if c[i] {
continue
}
gap = i - lastQuadSpecial
if isSquare(gap) {
sqrt := int(math.Sqrt(float64(gap)))
fmt.Printf("%7s %7s %6s %4d\n", commas(lastQuadSpecial), commas(i), commas(gap), sqrt)
lastQuadSpecial = i
count++
}
}
fmt.Println("\n", count+1, "such primes found.")
}</lang>
- Output:
Same as Wren example.
Julia
<lang julia>using Primes
function quadrat(N = 16000)
pmask = primesmask(1, N)
qprimes, lastn = [2], isqrt(N)
while (n = qprimes[end]) < N
for i in 1:lastn
q = n + i * i
if q > N
return qprimes
elseif pmask[q] # got next qprime
push!(qprimes, q)
break
end
end
end
end
println("Quadrat special primes < 16000:") foreach(p -> print(rpad(p[2], 6), p[1] % 10 == 0 ? "\n" : ""), enumerate(quadrat()))
</lang>
- Output:
Quadrat special primes < 16000: 2 3 7 11 47 83 227 263 587 911 947 983 1019 1163 1307 1451 1487 1523 1559 2459 3359 4259 4583 5483 5519 5843 5879 6203 6779 7103 7247 7283 7607 7643 8219 8363 10667 11243 11279 11423 12323 12647 12791 13367 13691 14591 14627 14771 15671
Phix
atom N=16000 sequence desc = split("linear quadratic cubic quartic quintic sextic septic") for p=2 to length(desc) do integer lastn = ceil(power(N,1/p)) sequence res = {2} bool done = false while not done do for n=1 to lastn do atom m = res[$] + power(n,p) if m>N then done = true exit elsif is_prime(m) then res &= m exit end if end for end while string r = join_by(apply(true,sprintf,{{"%,6d"},res}),1,p+5) printf(1,"Found %d %s special primes < %,d:\n%s\n",{length(res),desc[p],N,r}) end for
- Output:
Found 49 quadratic special primes < 16,000:
2 3 7 11 47 83 227
263 587 911 947 983 1,019 1,163
1,307 1,451 1,487 1,523 1,559 2,459 3,359
4,259 4,583 5,483 5,519 5,843 5,879 6,203
6,779 7,103 7,247 7,283 7,607 7,643 8,219
8,363 10,667 11,243 11,279 11,423 12,323 12,647
12,791 13,367 13,691 14,591 14,627 14,771 15,671
Found 23 cubic special primes < 16,000:
2 3 11 19 83 1,811 2,027 2,243
2,251 2,467 2,531 2,539 3,539 3,547 4,547 5,059
10,891 12,619 13,619 13,627 13,691 13,907 14,419
Found 3 quartic special primes < 16,000:
2 3 19
Found 2 quintic special primes < 16,000:
2 3
Found 4 sextic special primes < 16,000:
2 3 67 131
Found 3 septic special primes < 16,000:
2 3 131
Raku
<lang perl6>my @sqp = 2, -> $previous {
my $next;
for (1..∞).map: *² {
$next = $previous + $_;
last if $next.is-prime;
}
$next
} … *;
say "{+$_} matching numbers:\n", $_».fmt('%5d').batch(7).join: "\n" given
@sqp[^(@sqp.first: * > 16000, :k)];</lang>
- Output:
49 matching numbers:
2 3 7 11 47 83 227
263 587 911 947 983 1019 1163
1307 1451 1487 1523 1559 2459 3359
4259 4583 5483 5519 5843 5879 6203
6779 7103 7247 7283 7607 7643 8219
8363 10667 11243 11279 11423 12323 12647
12791 13367 13691 14591 14627 14771 15671
REXX
<lang rexx>/*REXX program finds the smallest primes such that the difference of successive terms */ /*─────────────────────────────────────────────────── are the smallest quadrat numbers. */ parse arg hi cols . /*obtain optional argument from the CL.*/ if hi== | hi=="," then hi= 16000 /* " " " " " " */ if cols== | cols=="," then cols= 10 /* " " " " " " */ call genP /*build array of semaphores for primes.*/ w= 10 /*width of a number in any column. */
@sqp= 'the smallest primes < ' commas(hi) " such that the" ,
'difference of successive terma are the smallest quadrat numbers'
if cols>0 then say ' index │'center(@sqp , 1 + cols*(w+1) ) if cols>0 then say '───────┼'center("" , 1 + cols*(w+1), '─') sqp= 0; idx= 1 /*initialize number of sqp and index.*/ op= 1 $= /*a list of nice primes (so far). */
do j=0 by 0
do k=1 until !.np; np= op + k*k /*find the next square + oldPrime.*/
if np>= hi then leave j /*Is newPrime too big? Then quit.*/
end /*k*/
sqp= sqp + 1 /*bump the number of sqp's. */
op= np /*assign the newPrime to the oldPrime*/
if cols==0 then iterate /*Build the list (to be shown later)? */
c= commas(np) /*maybe add commas to the number. */
$= $ right(c, max(w, length(c) ) ) /*add a nice prime ──► list, allow big#*/
if sqp//cols\==0 then iterate /*have we populated a line of output? */
say center(idx, 7)'│' substr($, 2); $= /*display what we have so far (cols). */
idx= idx + cols /*bump the index count for the output*/
end /*j*/
if $\== then say center(idx, 7)"│" substr($, 2) /*possible display residual output.*/ say say 'Found ' commas(sqp) " of " @sqp 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 ? /*──────────────────────────────────────────────────────────────────────────────────────*/ genP: !.= 0 /*placeholders for primes (semaphores).*/
@.1=2; @.2=3; @.3=5; @.4=7; @.5=11 /*define some low primes. */
!.2=1; !.3=1; !.5=1; !.7=1; !.11=1 /* " " " " flags. */
#=5; s.#= @.# **2 /*number of primes so far; prime². */
/* [↓] generate more primes ≤ high.*/
do j=@.#+2 by 2 to hi /*find odd primes from here on. */
parse var j -1 _; if _==5 then iterate /*J divisible by 5? (right dig)*/
if j// 3==0 then iterate /*" " " 3? */
if j// 7==0 then iterate /*" " " 7? */
/* [↑] the above five lines saves time*/
do k=5 while s.k<=j /* [↓] divide by the known odd primes.*/
if j // @.k == 0 then iterate j /*Is J ÷ X? Then not prime. ___ */
end /*k*/ /* [↑] only process numbers ≤ √ J */
#= #+1; @.#= j; s.#= j*j; !.j= 1 /*bump # of Ps; assign next P; P²; P# */
end /*j*/; return</lang>
- output when using the default inputs:
index │ the smallest primes < 16,000 such that the difference of successive terma are the smallest quadrat numbers ───────┼─────────────────────────────────────────────────────────────────────────────────────────────────────────────── 1 │ 2 3 7 11 47 83 227 263 587 911 11 │ 947 983 1,019 1,163 1,307 1,451 1,487 1,523 1,559 2,459 21 │ 3,359 4,259 4,583 5,483 5,519 5,843 5,879 6,203 6,779 7,103 31 │ 7,247 7,283 7,607 7,643 8,219 8,363 10,667 11,243 11,279 11,423 41 │ 12,323 12,647 12,791 13,367 13,691 14,591 14,627 14,771 15,671 Found 49 of the smallest primes < 16,000 such that the difference of successive terma are the smallest quadrat numbers
Ring
<lang ring>load "stdlib.ring"
/* Searching for the smallest prime gaps under a limit,
such that the difference of successive terms (gaps) is of the smallest degree. */
? "working..."
desc = split("na quadratic cubic quartic quintic sextic septic octic nonic decic"," ") limits = [1, 16000, 15000, 30000, 50000, 50000, 50000, 75000, 300000, 500000] for deg = 2 to len(desc)
Primes = []
limit = limits[deg]
oldPrime = 2
add(Primes, 2)
for n = 1 to sqrt(limit)
nextPrime = oldPrime + pow(n, deg)
if isprime(nextPrime)
n = 1
if nextPrime < limit add(Primes, nextPrime) ok
oldPrime = nextPrime
else
nextPrime = nextPrime - oldPrime
ok
if nextPrime > limit exit ok
next
? nl + desc[deg] + ":" + nl + " prime1 prime2 Gap Rt"
for n = 1 to Len(Primes) - 1
diff = Primes[n + 1] - Primes[n]
? sf(Primes[n], 7) + " " + sf(Primes[n+1], 7) + " " + sf(diff, 6) + " " + sf(floor(0.49 + pow(diff, 1 / deg)), 4)
next
? "Found " + Len(Primes) + " primes under " + limit + " for " + desc[deg] + " gaps."
next ? nl + "done..."
- a very plain string formatter, intended to even up columnar outputs
def sf x, y
s = string(x) l = len(s)
if l > y y = l ok
return substr(" ", 11 - y + l) + s</lang>
- Output:
working...
quadratic:
prime1 prime2 Gap Rt
2 3 1 1
3 7 4 2
7 11 4 2
11 47 36 6
47 83 36 6
83 227 144 12
227 263 36 6
263 587 324 18
587 911 324 18
911 947 36 6
947 983 36 6
983 1019 36 6
1019 1163 144 12
1163 1307 144 12
1307 1451 144 12
1451 1487 36 6
1487 1523 36 6
1523 1559 36 6
1559 2459 900 30
2459 3359 900 30
3359 4259 900 30
4259 4583 324 18
4583 5483 900 30
5483 5519 36 6
5519 5843 324 18
5843 5879 36 6
5879 6203 324 18
6203 6779 576 24
6779 7103 324 18
7103 7247 144 12
7247 7283 36 6
7283 7607 324 18
7607 7643 36 6
7643 8219 576 24
8219 8363 144 12
8363 10667 2304 48
10667 11243 576 24
11243 11279 36 6
11279 11423 144 12
11423 12323 900 30
12323 12647 324 18
12647 12791 144 12
12791 13367 576 24
13367 13691 324 18
13691 14591 900 30
14591 14627 36 6
14627 14771 144 12
14771 15671 900 30
Found 49 primes under 16000 for quadratic gaps.
cubic:
prime1 prime2 Gap Rt
2 3 1 1
3 11 8 2
11 19 8 2
19 83 64 4
83 1811 1728 12
1811 2027 216 6
2027 2243 216 6
2243 2251 8 2
2251 2467 216 6
2467 2531 64 4
2531 2539 8 2
2539 3539 1000 10
3539 3547 8 2
3547 4547 1000 10
4547 5059 512 8
5059 10891 5832 18
10891 12619 1728 12
12619 13619 1000 10
13619 13627 8 2
13627 13691 64 4
13691 13907 216 6
13907 14419 512 8
Found 23 primes under 15000 for cubic gaps.
quartic:
prime1 prime2 Gap Rt
2 3 1 1
3 19 16 2
Found 3 primes under 30000 for quartic gaps.
quintic:
prime1 prime2 Gap Rt
2 3 1 1
3 32771 32768 8
32771 32803 32 2
32803 33827 1024 4
33827 41603 7776 6
Found 6 primes under 50000 for quintic gaps.
sextic:
prime1 prime2 Gap Rt
2 3 1 1
3 67 64 2
67 131 64 2
Found 4 primes under 50000 for sextic gaps.
septic:
prime1 prime2 Gap Rt
2 3 1 1
3 131 128 2
Found 3 primes under 50000 for septic gaps.
octic:
prime1 prime2 Gap Rt
2 3 1 1
3 65539 65536 4
Found 3 primes under 75000 for octic gaps.
nonic:
prime1 prime2 Gap Rt
2 3 1 1
3 262147 262144 4
Found 3 primes under 300000 for nonic gaps.
decic:
prime1 prime2 Gap Rt
2 3 1 1
Found 2 primes under 500000 for decic gaps.
done...
Wren
<lang ecmascript>import "/math" for Int import "/fmt" for Fmt
var isSquare = Fn.new { |n|
var s = n.sqrt.floor return s*s == n
}
var primes = Int.primeSieve(15999) System.print("Quadrat special primes under 16,000:") System.print(" Prime1 Prime2 Gap Sqrt") var lastQuadSpecial = 3 var gap = 1 var count = 1 Fmt.print("$,7d $,7d $,6d $4d", 2, 3, 1, 1) for (p in primes.skip(2)) {
gap = p - lastQuadSpecial
if (isSquare.call(gap)) {
Fmt.print("$,7d $,7d $,6d $4d", lastQuadSpecial, p, gap, gap.sqrt)
lastQuadSpecial = p
count = count + 1
}
} System.print("\n%(count+1) such primes found.")</lang>
- Output:
Quadrat special primes under 16,000:
Prime1 Prime2 Gap Sqrt
2 3 1 1
3 7 4 2
7 11 4 2
11 47 36 6
47 83 36 6
83 227 144 12
227 263 36 6
263 587 324 18
587 911 324 18
911 947 36 6
947 983 36 6
983 1,019 36 6
1,019 1,163 144 12
1,163 1,307 144 12
1,307 1,451 144 12
1,451 1,487 36 6
1,487 1,523 36 6
1,523 1,559 36 6
1,559 2,459 900 30
2,459 3,359 900 30
3,359 4,259 900 30
4,259 4,583 324 18
4,583 5,483 900 30
5,483 5,519 36 6
5,519 5,843 324 18
5,843 5,879 36 6
5,879 6,203 324 18
6,203 6,779 576 24
6,779 7,103 324 18
7,103 7,247 144 12
7,247 7,283 36 6
7,283 7,607 324 18
7,607 7,643 36 6
7,643 8,219 576 24
8,219 8,363 144 12
8,363 10,667 2,304 48
10,667 11,243 576 24
11,243 11,279 36 6
11,279 11,423 144 12
11,423 12,323 900 30
12,323 12,647 324 18
12,647 12,791 144 12
12,791 13,367 576 24
13,367 13,691 324 18
13,691 14,591 900 30
14,591 14,627 36 6
14,627 14,771 144 12
14,771 15,671 900 30
49 such primes found.