Pell's equation

From Rosetta Code
Task
Pell's equation
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



ALGOL 68[edit]

Translation of: Sidef
Also tests for a trival solution only (if n is a perfect square only 1, 0 is solution).
Works with: ALGOL 68G version Any - tested with release 2.8.3.win32
BEGIN
# find solutions to Pell's eqauation: x^2 - ny^2 = 1 for integer x, y, n #
MODE BIGINT = LONG LONG INT;
MODE BIGPAIR = STRUCT( BIGINT v1, v2 );
PROC solve pell = ( INT n )BIGPAIR:
IF INT x = ENTIER( sqrt( n ) );
x * x = n
THEN
# n is a erfect square - no solution otheg than 1,0 #
BIGPAIR( 1, 0 )
ELSE
# there are non-trivial solutions #
INT y := x;
INT z := 1;
INT r := 2*x;
BIGPAIR e := BIGPAIR( 1, 0 );
BIGPAIR f := BIGPAIR( 0, 1 );
BIGINT a := 0;
BIGINT b := 0;
WHILE
y := (r*z - y);
z := ENTIER ((n - y*y) / z);
r := ENTIER ((x + y) / z);
e := BIGPAIR( v2 OF e, r * v2 OF e + v1 OF e );
f := BIGPAIR( v2 OF f, r * v2 OF f + v1 OF f );
a := (v2 OF e + x*v2 OF f);
b := v2 OF f;
a*a - n*b*b /= 1
DO SKIP OD;
BIGPAIR( a, b )
FI # solve pell # ;
# task test cases #
[]INT nv = (61, 109, 181, 277);
FOR i FROM LWB nv TO UPB nv DO
INT n = nv[ i ];
BIGPAIR r = solve pell(n);
print( ("x^2 - ", whole( n, -3 ), " * y^2 = 1 for x = ", whole( v1 OF r, -21), " and y = ", whole( v2 OF r, -21 ), newline ) )
OD
END
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

C#[edit]

Translation of: Sidef
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);
}
}
}
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

D[edit]

Translation of: C#
import std.bigint;
import std.math;
import std.stdio;
 
void fun(ref BigInt a, ref BigInt b, int c) {
auto t = a;
a = b;
b = b * c + t;
}
 
void solvePell(int n, ref BigInt a, ref BigInt b) {
int x = cast(int) sqrt(cast(real) n);
int y = x;
int z = 1;
int r = x << 1;
BigInt e1 = 1;
BigInt e2 = 0;
BigInt f1 = 0;
BigInt 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) {
return;
}
}
}
 
void main() {
BigInt x, y;
foreach(n; [61, 109, 181, 277]) {
solvePell(n, x, y);
writefln("x^2 - %3d * y^2 = 1 for x = %27d and y = %25d", n, x, y);
}
}
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

FreeBASIC[edit]

Translation of: Visual Basic .NET

for n = 277 the result is wrong, I do not know if you can represent such large numbers in 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
 
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[edit]

Translation of: Sidef
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)
}
}
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[edit]

Translation of: C#
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
 
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

Kotlin[edit]

Translation of: C#
import java.math.BigInteger
import kotlin.math.sqrt
 
class BIRef(var value: BigInteger) {
operator fun minus(b: BIRef): BIRef {
return BIRef(value - b.value)
}
 
operator fun times(b: BIRef): BIRef {
return BIRef(value * b.value)
}
 
override fun equals(other: Any?): Boolean {
if (this === other) return true
if (javaClass != other?.javaClass) return false
 
other as BIRef
 
if (value != other.value) return false
 
return true
}
 
override fun hashCode(): Int {
return value.hashCode()
}
 
override fun toString(): String {
return value.toString()
}
}
 
fun f(a: BIRef, b: BIRef, c: Int) {
val t = a.value
a.value = b.value
b.value = b.value * BigInteger.valueOf(c.toLong()) + t
}
 
fun solvePell(n: Int, a: BIRef, b: BIRef) {
val x = sqrt(n.toDouble()).toInt()
var y = x
var z = 1
var r = x shl 1
val e1 = BIRef(BigInteger.ONE)
val e2 = BIRef(BigInteger.ZERO)
val f1 = BIRef(BigInteger.ZERO)
val f2 = BIRef(BigInteger.ONE)
while (true) {
y = r * z - y
z = (n - y * y) / z
r = (x + y) / z
f(e1, e2, r)
f(f1, f2, r)
a.value = f2.value
b.value = e2.value
f(b, a, x)
if (a * a - BIRef(n.toBigInteger()) * b * b == BIRef(BigInteger.ONE)) {
return
}
}
}
 
fun main() {
val x = BIRef(BigInteger.ZERO)
val y = BIRef(BigInteger.ZERO)
intArrayOf(61, 109, 181, 277).forEach {
solvePell(it, x, y)
println("x^2 - %3d * y^2 = 1 for x = %,27d and y = %,25d".format(it, x.value, y.value))
}
}
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

Perl[edit]

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);
}
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[edit]

Works with: Rakudo version 2018.12
Translation of: Perl
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)».&comma;
}
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[edit]

Translation of: C#
Translation of: Go
Library: mpfr

This now ignores the nonsquare part of the task spec, returning {1,0}.

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
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

Python[edit]

Translation of: D
import math
 
def fun(a, b, c):
t = a[0]
a[0] = b[0]
b[0] = b[0] * c + t
 
def solvePell(n, a, b):
x = int(math.sqrt(n))
y = x
z = 1
r = x << 1
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[0] = f2[0]
b[0] = e2[0]
fun(b, a, x)
if a[0] * a[0] - n * b[0] * b[0] == 1:
return
 
x = [0]
y = [0]
for n in [61, 109, 181, 277]:
solvePell(n, x, y)
print("x^2 - %3d * y^2 = 1 for x = %27d and y = %25d" % (n, x[0], y[0]))
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

REXX[edit]

/*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
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[edit]

Translation of: Sidef
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)]}
 
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[edit]

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)
}
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[edit]

Translation of: Sidef
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
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[edit]

Library: GMP
GNU Multiple Precision Arithmetic Library
Translation of: Perl6
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);
}
}
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));
}
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