#*****************************************************************************80

#

## JACOBI_SYMBOL evaluates the Jacobi symbol (Q/P).

#

# Definition:

#

# If P is prime, then

#

# Jacobi Symbol (Q/P) = Legendre Symbol (Q/P)

#

# Else

#

# let P have the prime factorization

#

# P = Product ( 1 <= I <= N ) P(I)^E(I)

#

# Jacobi Symbol (Q/P) =

#

# Product ( 1 <= I <= N ) Legendre Symbol (Q/P(I))^E(I)

#

# Licensing:

#

# This code is distributed under the GNU LGPL license.

#

# Modified:

#

# 17 February 2015

#

# Author:

#

# John Burkardt

#

# Reference:

#

# Daniel Zwillinger,

# CRC Standard Mathematical Tables and Formulae,

# 30th Edition,

# CRC Press, 1996, pages 86-87.

#

# Parameters:

#

# Input, integer Q, an integer whose Jacobi symbol with

# respect to P is desired.

#

# Input, integer P, the number with respect to which the Jacobi

# symbol of Q is desired. P should be 2 or greater.

#

# Output, integer J, the Jacobi symbol (Q/P).

# Ordinarily, J will be -1, 0 or 1.

# -2, not enough factorization space.

# -3, an error during Legendre symbol calculation.

#

from i4_factor import i4_factor

from legendre_symbol import legendre_symbol

#

# P must be greater than 1.

#

if ( p <= 1 ):

print ( '' )

print ( 'JACOBI_SYMBOL - Fatal error!' )

print ( ' P must be greater than 1.' )

j = -2

return l

#

# Decompose P into factors of prime powers.

#

nfactor, factor, power, nleft = i4_factor ( p )

if ( nleft != 1 ):

print ( '' )

print ( 'JACOBI_SYMBOL - Fatal error!' )

print ( ' Not enough factorization space.' )

j = -2

return j

#

# Force Q to be nonnegative.

#

qq = q

while ( qq < 0 ):

qq = qq + p

#

# For each prime factor, compute the Legendre symbol, and

# multiply the Jacobi symbol by the appropriate factor.

#

j = 1

for i in range ( 0, nfactor ):

pp = factor[i]

l = legendre_symbol ( qq, pp )

if ( l < -1 ):

print ( '' )

print ( 'JACOBI_SYMBOL - Fatal error!' )

print ( ' Error during Legendre symbol calculation.' )

j = -3

j = j * l ** power[i]

#*****************************************************************************80

#

## JACOBI_SYMBOL_TEST_TEST tests JACOBI_SYMBOL_TEST.

#

# Licensing:

#

# This code is distributed under the GNU LGPL license.

#

# Modified:

#

# 17 February 2015

#

# Author:

#

# John Burkardt

#

import numpy as np

import platform

ntest = 4

ptest = [ 3, 9, 10, 12 ]

print ( '' )

print ( 'JACOBI_SYMBOL_TEST' )

print ( ' Python version: %s' % ( platform.python_version ( ) ) )

print ( ' JACOBI_SYMBOL computes the Jacobi symbol' )

print ( ' (Q/P) which records whether Q is' )

print ( ' a quadratic residue modulo the number P.' )

for i in range ( 0, ntest ):

p = ptest[i]

print ( '' )

print ( ' Jacobi Symbols for P = %d' % ( p ) )

print ( '' )

for q in range ( 0, p + 1 ):

l = jacobi_symbol ( q, p )

print ( ' %6d %6d %6d' % ( p, q, l ) )

#

# Terminate.

#

print ( '' )

print ( 'JACOBI_SYMBOL_TEST' )

print ( ' Normal end of execution.' )