def pepins_test(k, Fn):
leg = legendre_symbol(k, Fn)
jac = jacobi(k, Fn, verbose=False)
print("Fn\t\t:\t{}".format(Fn))
print("Legendre Symbol\t:\t{}".format(leg))
print("Jacobi Symbol\t:\t{}".format(jac))
if leg == jac:
return True
else:
return False
def conference_inverse ( n ):
#*****************************************************************************80
#
## CONFERENCE_INVERSE returns the inverse of the CONFERENCE matrix.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 19 April 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer N, the order of the matrix.
#
# Output, real CONFERENCE_INVERSE[N,N], the matrix.
#
import numpy as np
from legendre_symbol import legendre_symbol
a = np.zeros ( ( n, n ) )
for j in range ( 0, n ):
for i in range ( 0, n ):
if ( i == 0 and j == 0 ):
a[i,j] = 0.0
elif ( i == 0 ):
a[i,j] = 1.0
elif ( j == 0 ):
if ( ( ( n - 1 ) % 4 ) == 1 ):
a[i,j] = 1.0
else:
a[i,j] = - 1.0
else:
l = legendre_symbol ( i - j, n - 1 )
a[i,j] = float ( l )
if ( ( ( n - 1 ) % 4 ) == 3 ):
for i in range ( 0, n ):
for j in range ( 0, n ):
a[i,j] = - a[i,j]
if ( 1 < n ):
for i in range ( 0, n ):
for j in range ( 0, n ):
a[i,j] = a[i,j] / float ( n - 1 )
return a
def conference_inverse(n):
#*****************************************************************************80
#
## CONFERENCE_INVERSE returns the inverse of the CONFERENCE matrix.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 19 April 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer N, the order of the matrix.
#
# Output, real CONFERENCE_INVERSE[N,N], the matrix.
#
import numpy as np
from legendre_symbol import legendre_symbol
a = np.zeros((n, n))
for j in range(0, n):
for i in range(0, n):
if (i == 0 and j == 0):
a[i, j] = 0.0
elif (i == 0):
a[i, j] = 1.0
elif (j == 0):
if (((n - 1) % 4) == 1):
a[i, j] = 1.0
else:
a[i, j] = -1.0
else:
l = legendre_symbol(i - j, n - 1)
a[i, j] = float(l)
if (((n - 1) % 4) == 3):
for i in range(0, n):
for j in range(0, n):
a[i, j] = -a[i, j]
if (1 < n):
for i in range(0, n):
for j in range(0, n):
a[i, j] = a[i, j] / float(n - 1)
return a
def conference ( n ):
#*****************************************************************************80
#
## CONFERENCE returns the CONFERENCE matrix.
#
# Discussion:
#
# A conference matrix is a square matrix A of order N, with a zero
# diagonal, and only 1's and -1's on the offdiagonal, with the property
# that:
#
# A * A' = (N-1) * I.
#
# The algorithm employed here is only valid when N - 1
# is an odd prime, or a power of an odd prime.
#
# Conference matrices have a relationship with Hadamard matrices:
#
# If mod ( P, 4 ) == 3, A is antisymmetric, and
# I + A is hadamard;
# Else A is symmetric, and
# ( I + A, - I + A )
# ( - I + A, - I - A) is Hadamard.
#
# Example:
#
# N = 8
#
# 0 1 1 1 1 1 1 1
# -1 0 -1 -1 1 -1 1 1
# -1 1 0 -1 -1 1 -1 1
# -1 1 1 0 -1 -1 1 -1
# -1 -1 1 1 0 -1 -1 1
# -1 1 -1 1 1 0 -1 -1
# -1 -1 1 -1 1 1 0 -1
# -1 -1 -1 1 -1 1 1 0
#
# Properties:
#
# If N-1 is prime, then A[2:N,2:N] is a circulant matrix.
#
# If N-1 = 1 mod 4, then A is symmetric.
#
# If N-1 = 3 mod 4, then A is antisymmetric.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 19 April 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer N, the order of the matrix. N-1 must be an
# odd prime, or a power of an odd prime.
#
# Output, real A(N,N), the matrix.
#
import numpy as np
from legendre_symbol import legendre_symbol
a = np.zeros ( ( n, n ) )
for j in range ( 0, n ):
for i in range ( 0, n ):
if ( i == 0 and j == 0 ):
a[i,j] = 0.0
elif ( i == 0 ):
a[i,j] = 1.0
elif ( j == 0 ):
if ( ( ( n - 1 ) % 4 ) == 1 ):
a[i,j] = 1.0
else:
a[i,j] = -1.0
else:
l = legendre_symbol ( i - j, n - 1 )
a[i,j] = float ( l )
return a
def jacobi_symbol(q, p):
#*****************************************************************************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]
return j
def jacobi_symbol ( q, p ):
#*****************************************************************************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 1
#
# Decompose P into factors of prime powers.
#
nfactor, factor, power, nleft = pfdic(p)#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]
return j
def conference(n):
#*****************************************************************************80
#
## CONFERENCE returns the CONFERENCE matrix.
#
# Discussion:
#
# A conference matrix is a square matrix A of order N, with a zero
# diagonal, and only 1's and -1's on the offdiagonal, with the property
# that:
#
# A * A' = (N-1) * I.
#
# The algorithm employed here is only valid when N - 1
# is an odd prime, or a power of an odd prime.
#
# Conference matrices have a relationship with Hadamard matrices:
#
# If mod ( P, 4 ) == 3, A is antisymmetric, and
# I + A is hadamard;
# Else A is symmetric, and
# ( I + A, - I + A )
# ( - I + A, - I - A) is Hadamard.
#
# Example:
#
# N = 8
#
# 0 1 1 1 1 1 1 1
# -1 0 -1 -1 1 -1 1 1
# -1 1 0 -1 -1 1 -1 1
# -1 1 1 0 -1 -1 1 -1
# -1 -1 1 1 0 -1 -1 1
# -1 1 -1 1 1 0 -1 -1
# -1 -1 1 -1 1 1 0 -1
# -1 -1 -1 1 -1 1 1 0
#
# Properties:
#
# If N-1 is prime, then A[2:N,2:N] is a circulant matrix.
#
# If N-1 = 1 mod 4, then A is symmetric.
#
# If N-1 = 3 mod 4, then A is antisymmetric.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 19 April 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer N, the order of the matrix. N-1 must be an
# odd prime, or a power of an odd prime.
#
# Output, real A(N,N), the matrix.
#
import numpy as np
from legendre_symbol import legendre_symbol
a = np.zeros((n, n))
for j in range(0, n):
for i in range(0, n):
if (i == 0 and j == 0):
a[i, j] = 0.0
elif (i == 0):
a[i, j] = 1.0
elif (j == 0):
if (((n - 1) % 4) == 1):
a[i, j] = 1.0
else:
a[i, j] = -1.0
else:
l = legendre_symbol(i - j, n - 1)
a[i, j] = float(l)
return a