Pell's equation: Difference between revisions
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=={{header|C#}}== |
=={{header|C#}}== |
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{{trans| |
{{trans|Sidef}} |
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<lang csharp>using System; |
<lang csharp>using System; |
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using System.Numerics; |
using System.Numerics; |
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=={{header|Visual Basic .NET}}== |
=={{header|Visual Basic .NET}}== |
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{{trans| |
{{trans|Sidef}} |
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<lang vbnet>Imports System.Numerics |
<lang vbnet>Imports System.Numerics |
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Revision as of 03:50, 3 February 2019
Pell's equation is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
Pell's equation is a Diophantine equation of the form
- x2 - ny2 = 1
with integer solutions for x and y, where n is a given nonsquare 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
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
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>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 -> $n {
my ($x, $y) = pell($n); printf("x² - %3dy² = 1 for x = %-21s and y = %s\n", $n, $x, $y);
}</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
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