NAME
    jacobi - Jacobi symbol function

SYNOPSIS
    jacobi(x, y)

TYPES
    x		integer
    y		integer

    return	1, -1, or 0

DESCRIPTION
    If y is a positive odd prime and x is an integer not divisible
    by y, jacobi(x,y) returns the Legendre symbol function, usually
    denoted by (x/y) as if x/y were a fraction; this has the value
    1 or -1 according as x is or is not a quadratic residue modulo y.
    x is a quadratic residue modulo y if for some integer u,
    x = u^2 (mod y); if for all integers u, x != u^2 (mod y), x
    is said to be a quadratic nonresidue modulo y.

    If y is a positive odd prime and x is divisible by y, jacobi(x,y)
    returns the value 1.  (This differs from the zero value usually
    given in number theory books for (x/y) when x and y
    are not relatively prime.)
    assigned to (x/y) O

    If y is an odd positive integer equal to p_1 * p_2 * ... * p_k,
    where the p_i are primes, not necessarily distinct, the
    jacobi symbol function is given by

		jacobi(x,y)  =	(x/p_1) * (x/p_2) * ... * (x/p_k).

    where the functions on the right are Legendre symbol functions.

    This is also often usually by (x/y).

    If jacobi(x,y) = -1, then x is a quadratic nonresidue modulo y.
    Equivalently, if x is a quadratic residue modulo y, then
    jacobi(x,y) = 1.

    If jacobi(x,y) = 1 and y is composite, x may be either a quadratic
    residue or a quadratic nonresidue modulo y.

    If y is even or negative, jacobi(x,y) as defined by calc returns
    the value 0.

EXAMPLE
    ; print jacobi(2,3), jacobi(2,5), jacobi(2,15)
    -1 -1 1

    ; print jacobi(80,199)
    1

LIMITS
    none

LINK LIBRARY
    NUMBER *qjacobi(NUMBER *x, NUMBER *y)
    FLAG zjacobi(ZVALUE z1, ZVALUE z2)

SEE ALSO

## Copyright (C) 1999  Landon Curt Noll
##
## 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.
##
## @(#) $Revision: 30.1 $
## @(#) $Id: jacobi,v 30.1 2007/03/16 11:10:42 chongo Exp $
## @(#) $Source: /usr/local/src/bin/calc/help/RCS/jacobi,v $
##
## Under source code control:	1995/12/18 12:34:57
## File existed as early as:	1995
##
## chongo <was here> /\oo/\	http://www.isthe.com/chongo/
## Share and enjoy!  :-)	http://www.isthe.com/chongo/tech/comp/calc/
