Pell's equation: Difference between revisions
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{{task|Mathematics}} |
{{task|Mathematics}} |
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'''Pell's equation''' is a Diophantine equation of the form: |
'''Pell's equation''' (also called the '''Pell–Fermat''' equation) is a Diophantine equation of the form: |
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:::::: <big> <b> x<sup>2</sup> - ny<sup>2</sup> = 1 </b> </big> |
:::::: <big> <b> x<sup>2</sup> - ny<sup>2</sup> = 1 </b> </big> |
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Revision as of 21:30, 19 June 2019
You are encouraged to solve this task according to the task description, using any language you may know.
Pell's equation (also called the Pell–Fermat equation) is a Diophantine equation of the form:
- x2 - ny2 = 1
with integer solutions for x and y, where n is a given non-square positive integer.
- Task requirements
-
- find the smallest solution in positive integers to Pell's equation for n = {61, 109, 181, 277}.
- See also
-
- Wikipedia entry: Pell's equation.
C#
<lang csharp>using System; using System.Numerics;
static class Program {
static void Fun(ref BigInteger a, ref BigInteger b, int c)
{
BigInteger t = a; a = b; b = b * c + t;
}
static void SolvePell(int n, ref BigInteger a, ref BigInteger b)
{
int x = (int)Math.Sqrt(n), y = x, z = 1, r = x << 1;
BigInteger e1 = 1, e2 = 0, f1 = 0, f2 = 1;
while (true)
{
y = r * z - y; z = (n - y * y) / z; r = (x + y) / z;
Fun(ref e1, ref e2, r); Fun(ref f1, ref f2, r); a = f2; b = e2; Fun(ref b, ref a, x);
if (a * a - n * b * b == 1) return;
}
}
static void Main()
{
BigInteger x, y; foreach (int n in new[] { 61, 109, 181, 277 })
{
SolvePell(n, ref x, ref y);
Console.WriteLine("x^2 - {0,3} * y^2 = 1 for x = {1,27:n0} and y = {2,25:n0}", n, x, y);
}
}
}</lang>
- Output:
x^2 - 61 * y^2 = 1 for x = 1,766,319,049 and y = 226,153,980 x^2 - 109 * y^2 = 1 for x = 158,070,671,986,249 and y = 15,140,424,455,100 x^2 - 181 * y^2 = 1 for x = 2,469,645,423,824,185,801 and y = 183,567,298,683,461,940 x^2 - 277 * y^2 = 1 for x = 159,150,073,798,980,475,849 and y = 9,562,401,173,878,027,020
FreeBASIC
for n = 277 the result is wrong, I do not know if you can represent such large numbers in FreeBasic! <lang freebasic> Sub Fun(Byref a As LongInt, Byref b As LongInt, c As Integer)
Dim As LongInt t t = a : a = b : b = b * c + t
End Sub
Sub SolvePell(n As Integer, Byref a As LongInt, Byref b As LongInt)
Dim As Integer z, r
Dim As LongInt x, y, e1, e2, f1, f2
x = Sqr(n) : y = x : z = 1 : r = 2 * x
e1 = 1 : e2 = 0 : f1 = 0 : f2 = 1
While True
y = r * z - y : z = (n - y * y) / z : r = (x + y) / z
Fun(e1, e2, r) : Fun(f1, f2, r) : a = f2 : b = e2 : Fun(b, a, x)
If a * a - n * b * b = 1 Then Exit Sub
Wend
End Sub
Dim As Integer i Dim As LongInt x, y Dim As Integer n(0 To 3) = {61, 109, 181, 277} For i = 0 To 3 n In {61, 109, 181, 277}
SolvePell(n(i), x, y) Print Using "x^2 - ### * y^2 = 1 for x = ##################### and y = #####################"; n(i); x; y
Next i </lang>
- Output:
x^2 - 61 * y^2 = 1 for x = 1766319049 and y = 226153980 x^2 - 109 * y^2 = 1 for x = 158070671986249 and y = 15140424455100 x^2 - 181 * y^2 = 1 for x = 2469645423824185801 and y = 183567298683461940 x^2 - 277 * y^2 = 1 for x = -6870622864405488695 and y = -8884342899831524596
Go
<lang go>package main
import (
"fmt" "math/big"
)
var big1 = new(big.Int).SetUint64(1)
func solvePell(nn uint64) (*big.Int, *big.Int) {
n := new(big.Int).SetUint64(nn) x := new(big.Int).Set(n) x.Sqrt(x) y := new(big.Int).Set(x) z := new(big.Int).SetUint64(1) r := new(big.Int).Lsh(x, 1)
e1 := new(big.Int).SetUint64(1) e2 := new(big.Int) f1 := new(big.Int) f2 := new(big.Int).SetUint64(1)
t := new(big.Int)
u := new(big.Int)
a := new(big.Int)
b := new(big.Int)
for {
t.Mul(r, z)
y.Sub(t, y)
t.Mul(y, y)
t.Sub(n, t)
z.Quo(t, z)
t.Add(x, y)
r.Quo(t, z)
u.Set(e1)
e1.Set(e2)
t.Mul(r, e2)
e2.Add(t, u)
u.Set(f1)
f1.Set(f2)
t.Mul(r, f2)
f2.Add(t, u)
t.Mul(x, f2)
a.Add(e2, t)
b.Set(f2)
t.Mul(a, a)
u.Mul(n, b)
u.Mul(u, b)
t.Sub(t, u)
if t.Cmp(big1) == 0 {
return a, b
}
}
}
func main() {
ns := []uint64{61, 109, 181, 277}
for _, n := range ns {
x, y := solvePell(n)
fmt.Printf("x^2 - %3d*y^2 = 1 for x = %-21s and y = %s\n", n, x, y)
}
}</lang>
- Output:
x^2 - 61*y^2 = 1 for x = 1766319049 and y = 226153980 x^2 - 109*y^2 = 1 for x = 158070671986249 and y = 15140424455100 x^2 - 181*y^2 = 1 for x = 2469645423824185801 and y = 183567298683461940 x^2 - 277*y^2 = 1 for x = 159150073798980475849 and y = 9562401173878027020
Julia
<lang julia>function pell(n)
x = BigInt(floor(sqrt(n)))
y, z, r = x, BigInt(1), x << 1
e1, e2, f1, f2 = BigInt(1), BigInt(0), BigInt(0), BigInt(1)
while true
y = r * z - y
z = div(n - y * y, z)
r = div(x + y, z)
e1, e2 = e2, e2 * r + e1
f1, f2 = f2, f2 * r + f1
a, b = f2, e2
b, a = a, a * x + b
if a * a - n * b * b == 1
return a, b
end
end
end
for target in BigInt[61, 109, 181, 277]
x, y = pell(target)
println("x\u00b2 - $target", "y\u00b2 = 1 for x = $x and y = $y")
end
</lang>
- Output:
x² - 61y² = 1 for x = 1766319049 and y = 226153980 x² - 109y² = 1 for x = 158070671986249 and y = 15140424455100 x² - 181y² = 1 for x = 2469645423824185801 and y = 183567298683461940 x² - 277y² = 1 for x = 159150073798980475849 and y = 9562401173878027020
Perl
<lang perl>sub solve_pell {
my ($n) = @_;
use bigint try => 'GMP';
my $x = int(sqrt($n)); my $y = $x; my $z = 1; my $r = 2 * $x;
my ($e1, $e2) = (1, 0); my ($f1, $f2) = (0, 1);
for (; ;) {
$y = $r * $z - $y;
$z = int(($n - $y * $y) / $z);
$r = int(($x + $y) / $z);
($e1, $e2) = ($e2, $r * $e2 + $e1);
($f1, $f2) = ($f2, $r * $f2 + $f1);
my $A = $e2 + $x * $f2;
my $B = $f2;
if ($A**2 - $n * $B**2 == 1) {
return ($A, $B);
}
}
}
foreach my $n (61, 109, 181, 277) {
my ($x, $y) = solve_pell($n);
printf("x^2 - %3d*y^2 = 1 for x = %-21s and y = %s\n", $n, $x, $y);
}</lang>
- Output:
x^2 - 61*y^2 = 1 for x = 1766319049 and y = 226153980 x^2 - 109*y^2 = 1 for x = 158070671986249 and y = 15140424455100 x^2 - 181*y^2 = 1 for x = 2469645423824185801 and y = 183567298683461940 x^2 - 277*y^2 = 1 for x = 159150073798980475849 and y = 9562401173878027020
Perl 6
<lang perl6>use Lingua::EN::Numbers;
sub pell (Int $n) {
my $y = my $x = Int(sqrt $n); my $z = 1; my $r = 2 * $x;
my ($e1, $e2) = (1, 0); my ($f1, $f2) = (0, 1);
loop {
$y = $r * $z - $y;
$z = Int(($n - $y²) / $z);
$r = Int(($x + $y) / $z);
($e1, $e2) = ($e2, $r * $e2 + $e1);
($f1, $f2) = ($f2, $r * $f2 + $f1);
my $A = $e2 + $x * $f2;
my $B = $f2;
if ($A² - $n * $B² == 1) {
return ($A, $B);
}
}
}
for 61, 109, 181, 277, 8941 -> $n {
next if $n.sqrt.narrow ~~ Int; my ($x, $y) = pell($n); printf "x² - %sy² = 1 for:\n\tx = %s\n\ty = %s\n\n", $n, |($x, $y)».,
}</lang>
- Output:
x² - 61y² = 1 for: x = 1,766,319,049 y = 226,153,980 x² - 109y² = 1 for: x = 158,070,671,986,249 y = 15,140,424,455,100 x² - 181y² = 1 for: x = 2,469,645,423,824,185,801 y = 183,567,298,683,461,940 x² - 277y² = 1 for: x = 159,150,073,798,980,475,849 y = 9,562,401,173,878,027,020 x² - 8941y² = 1 for: x = 2,565,007,112,872,132,129,669,406,439,503,954,211,359,492,684,749,762,901,360,167,370,740,763,715,001,557,789,090,674,216,330,243,703,833,040,774,221,628,256,858,633,287,876,949,448,689,668,281,446,637,464,359,482,677,366,420,261,407,112,316,649,010,675,881,349,744,201 y = 27,126,610,172,119,035,540,864,542,981,075,550,089,190,381,938,849,116,323,732,855,930,990,771,728,447,597,698,969,628,164,719,475,714,805,646,913,222,890,277,024,408,337,458,564,351,161,990,641,948,210,581,361,708,373,955,113,191,451,102,494,265,278,824,127,994,180
Phix
This now ignores the nonsquare part of the task spec, returning {1,0}. <lang Phix>include mpfr.e
procedure fun(mpz a,b,t, integer c) -- {a,b} = {b,c*b+a} (and t gets trashed)
mpz_set(t,a) mpz_set(a,b) mpz_mul_si(b,b,c) mpz_add(b,b,t)
end procedure
function SolvePell(integer n) integer x = floor(sqrt(n)), y = x, z = 1, r = x*2 mpz e1 = mpz_init(1), e2 = mpz_init(),
f1 = mpz_init(), f2 = mpz_init(1),
t = mpz_init(0), u = mpz_init(),
a = mpz_init(1), b = mpz_init(0)
if x*x!=n then
while mpz_cmp_si(t,1)!=0 do
y = r*z - y
z = floor((n-y*y)/z)
r = floor((x+y)/z)
fun(e1,e2,t,r) -- {e1,e2} = {e2,r*e2+e1}
fun(f1,f2,t,r) -- {f1,f2} = {f2,r*r2+f1}
mpz_set(a,f2)
mpz_set(b,e2)
fun(b,a,t,x) -- {b,a} = {f2,x*f2+e2}
mpz_mul(t,a,a)
mpz_mul_si(u,b,n)
mpz_mul(u,u,b)
mpz_sub(t,t,u) -- t = a^2-n*b^2
end while
end if
return {a, b}
end function
sequence ns = {4, 61, 109, 181, 277, 8941} for i=1 to length(ns) do
integer n = ns[i]
mpz {x, y} = SolvePell(n)
string xs = mpz_get_str(x,comma_fill:=true),
ys = mpz_get_str(y,comma_fill:=true)
printf(1,"x^2 - %3d*y^2 = 1 for x = %27s and y = %25s\n", {n, xs, ys})
end for</lang>
- Output:
x^2 - 4*y^2 = 1 for x = 1 and y = 0
x^2 - 61*y^2 = 1 for x = 1,766,319,049 and y = 226,153,980
x^2 - 109*y^2 = 1 for x = 158,070,671,986,249 and y = 15,140,424,455,100
x^2 - 181*y^2 = 1 for x = 2,469,645,423,824,185,801 and y = 183,567,298,683,461,940
x^2 - 277*y^2 = 1 for x = 159,150,073,798,980,475,849 and y = 9,562,401,173,878,027,020
x^2 - 8941*y^2 = 1 for x = 2,565,007,112,872,132,129,669,406,439,503,954,211,359,492,684,749,762,
901,360,167,370,740,763,715,001,557,789,090,674,216,330,243,703,833,
040,774,221,628,256,858,633,287,876,949,448,689,668,281,446,637,464,
359,482,677,366,420,261,407,112,316,649,010,675,881,349,744,201
and y = 27,126,610,172,119,035,540,864,542,981,075,550,089,190,381,938,849,116,
323,732,855,930,990,771,728,447,597,698,969,628,164,719,475,714,805,
646,913,222,890,277,024,408,337,458,564,351,161,990,641,948,210,581,
361,708,373,955,113,191,451,102,494,265,278,824,127,994,180
REXX
<lang rexx>/*REXX program to solve Pell's equation for the smallest solution of positive integers. */ numeric digits 2200 /*ensure enough decimal digs for answer*/ parse arg $ /*obtain optional arguments from the CL*/ if $== | $=="," then $= 61 109 181 277 /*Not specified? Then use the defaults*/ d= 22 /*used for aligning the output numbers.*/
do j=1 for words($); #= word($, j) /*process all the numbers in the list. */
parse value pells(#) with x y /*extract the two values of X and Y.*/
say 'x^2 -'right(#,max(4,length(#))) "* y^2 == 1 when x="right(x, max(d,length(x))),
' and y='right(y, max(d,length(y)))
end /*j*/
exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ floor: procedure; parse arg x; _= x % 1; return _ - (x < 0) * (x \= _) /*──────────────────────────────────────────────────────────────────────────────────────*/ iSqrt: procedure; parse arg x; r= 0; q= 1; do while q<=x; q= q * 4; end
do while q>1; q= q%4; _= x-r-q; r= r%2; if _>=0 then do; x= _; r= r+q; end; end
return r /*R: is the integer square root of X. */
/*──────────────────────────────────────────────────────────────────────────────────────*/ pells: procedure; parse arg n; x= iSqrt(n); y=x /*obtain arg; obtain integer sqrt of N*/
parse value 1 0 with e1 e2 1 f2 f1 /*assign values for: E1, E2, and F2, F1*/
z= 1; r= x + x
do until ( (e2 + x*f2)**2 - n*f2*f2) == 1
y= r*z - y
z= floor( (n - y*y) / z)
r= floor( (x + y ) / z)
parse value e2 r*e2 + e1 with e1 e2
parse value f2 r*f2 + f1 with f1 f2
end /*until*/
return e2 + x * f2 f2</lang>
- output when using the default inputs:
x^2 - 61 * y^2 == 1 when x= 1766319049 and y= 226153980 x^2 - 109 * y^2 == 1 when x= 158070671986249 and y= 15140424455100 x^2 - 181 * y^2 == 1 when x= 2469645423824185801 and y= 183567298683461940 x^2 - 277 * y^2 == 1 when x= 159150073798980475849 and y= 9562401173878027020
Ruby
<lang ruby>def solve_pell(n)
x = Integer.sqrt(n) y = x z = 1 r = 2*x e1, e2 = 1, 0 f1, f2 = 0, 1
loop do y = r*z - y z = (n - y*y) / z r = (x + y) / z e1, e2 = e2, r*e2 + e1 f1, f2 = f2, r*f2 + f1 a, b = e2 + x*f2, f2 break a, b if a*a - n*b*b == 1 end
end
[61, 109, 181, 277].each {|n| puts "x*x - %3s*y*y = 1 for x = %-21s and y = %s" % [n, *solve_pell(n)]} </lang>
- Output:
x*x - 61*y*y = 1 for x = 1766319049 and y = 226153980 x*x - 109*y*y = 1 for x = 158070671986249 and y = 15140424455100 x*x - 181*y*y = 1 for x = 2469645423824185801 and y = 183567298683461940 x*x - 277*y*y = 1 for x = 159150073798980475849 and y = 9562401173878027020
Sidef
<lang ruby>func solve_pell(n) {
var x = n.isqrt var y = x var z = 1 var r = 2*x
var (e1, e2) = (1, 0) var (f1, f2) = (0, 1)
loop {
y = (r*z - y)
z = floor((n - y*y) / z)
r = floor((x + y) / z)
(e1, e2) = (e2, r*e2 + e1)
(f1, f2) = (f2, r*f2 + f1)
var A = (e2 + x*f2)
var B = f2
if (A**2 - n*B**2 == 1) {
return (A, B)
}
}
}
for n in [61, 109, 181, 277] {
var (x, y) = solve_pell(n)
printf("x^2 - %3d*y^2 = 1 for x = %-21s and y = %s\n", n, x, y)
}</lang>
- Output:
x^2 - 61*y^2 = 1 for x = 1766319049 and y = 226153980 x^2 - 109*y^2 = 1 for x = 158070671986249 and y = 15140424455100 x^2 - 181*y^2 = 1 for x = 2469645423824185801 and y = 183567298683461940 x^2 - 277*y^2 = 1 for x = 159150073798980475849 and y = 9562401173878027020
Visual Basic .NET
<lang vbnet>Imports System.Numerics
Module Module1
Sub Fun(ByRef a As BigInteger, ByRef b As BigInteger, c As Integer)
Dim t As BigInteger = a : a = b : b = b * c + t
End Sub
Sub SolvePell(n As Integer, ByRef a As BigInteger, ByRef b As BigInteger)
Dim x As Integer = Math.Sqrt(n), y As Integer = x, z As Integer = 1, r As Integer = x << 1,
e1 As BigInteger = 1, e2 As BigInteger = 0, f1 As BigInteger = 0, f2 As BigInteger = 1
While True
y = r * z - y : z = (n - y * y) / z : r = (x + y) / z
Fun(e1, e2, r) : Fun(f1, f2, r) : a = f2 : b = e2 : Fun(b, a, x)
If a * a - n * b * b = 1 Then Exit Sub
End While
End Sub
Sub Main()
Dim x As BigInteger, y As BigInteger
For Each n As Integer In {61, 109, 181, 277}
SolvePell(n, x, y)
Console.WriteLine("x^2 - {0,3} * y^2 = 1 for x = {1,27:n0} and y = {2,25:n0}", n, x, y)
Next
End Sub
End Module</lang>
- Output:
x^2 - 61 * y^2 = 1 for x = 1,766,319,049 and y = 226,153,980 x^2 - 109 * y^2 = 1 for x = 158,070,671,986,249 and y = 15,140,424,455,100 x^2 - 181 * y^2 = 1 for x = 2,469,645,423,824,185,801 and y = 183,567,298,683,461,940 x^2 - 277 * y^2 = 1 for x = 159,150,073,798,980,475,849 and y = 9,562,401,173,878,027,020
zkl
GNU Multiple Precision Arithmetic Library
<lang zkl>var [const] BI=Import("zklBigNum"); // libGMP
fcn solve_pell(n){
x,y,z,r := BI(n).root(2), x.copy(), BI(1), x*2;
e1,e2, f1,f2 := BI(1), BI(0), BI(0), BI(1);
reg t; // a,b = c,d is a=c; b=d
do(30_000){ // throttle this in case of screw up
y,z,r = (r*z - y), (n - y*y)/z, (x + y)/z;
t,e2,e1 = e2, r*e2 + e1, t;
t,f2,f1 = f2, r*f2 + f1, t;
A,B := e2 + x*f2, f2;
if (A*A - B*B*n == 1) return(A,B); }
}</lang> <lang zkl>foreach n in (T(61, 109, 181, 277)){
x,y:=solve_pell(n);
println("x^2 - %3d*y^2 = 1 for x = %-21d and y = %d".fmt(n,x,y));
}</lang>
- Output:
x^2 - 61*y^2 = 1 for x = 1766319049 and y = 226153980 x^2 - 109*y^2 = 1 for x = 158070671986249 and y = 15140424455100 x^2 - 181*y^2 = 1 for x = 2469645423824185801 and y = 183567298683461940 x^2 - 277*y^2 = 1 for x = 159150073798980475849 and y = 9562401173878027020