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/*
* pell - solve Pell's equation
*
* Copyright (C) 1999 David I. Bell
*
* Calc is open software; you can redistribute it and/or modify it under
* the terms of the version 2.1 of the GNU Lesser General Public License
* as published by the Free Software Foundation.
*
* Calc is distributed in the hope that it will be useful, but WITHOUT
* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
* or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General
* Public License for more details.
*
* A copy of version 2.1 of the GNU Lesser General Public License is
* distributed with calc under the filename COPYING-LGPL. You should have
* received a copy with calc; if not, write to Free Software Foundation, Inc.
* 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
*
* Under source code control: 1990/02/15 01:50:34
* File existed as early as: before 1990
*
* Share and enjoy! :-) http://www.isthe.com/chongo/tech/comp/calc/
*/
/*
* Solve Pell's equation; Returns the solution X to: X^2 - D * Y^2 = 1.
* Type the solution to pells equation for a particular D.
*/
define pell(D)
{
local X, Y;
X = pellx(D);
if (isnull(X)) {
print "D=":D:" is square";
return;
}
Y = isqrt((X^2 - 1) / D);
print X : "^2 - " : D : "*" : Y : "^2 = " : X^2 - D*Y^2;
}
/*
* Function to solve Pell's equation
* Returns the solution X to:
* X^2 - D * Y^2 = 1
*/
define pellx(D)
{
local R, Rp, U, Up, V, Vp, A, T, Q1, Q2, n;
local mat ans[2,2];
local mat tmp[2,2];
R = isqrt(D);
Vp = D - R^2;
if (Vp == 0)
return;
Rp = R + R;
U = Rp;
Up = U;
V = 1;
A = 0;
n = 0;
ans[0,0] = 1;
ans[1,1] = 1;
tmp[0,1] = 1;
tmp[1,0] = 1;
do {
T = V;
V = A * (Up - U) + Vp;
Vp = T;
A = U // V;
Up = U;
U = Rp - U % V;
tmp[0,0] = A;
ans *= tmp;
n++;
} while (A != Rp);
Q2 = ans[[1]];
Q1 = isqrt(Q2^2 * D + 1);
if (isodd(n)) {
T = Q1^2 + D * Q2^2;
Q2 = Q1 * Q2 * 2;
Q1 = T;
}
return Q1;
}