def semiCompositeIdempotentPartitions(n, factor_list):
partitions_of_n = partitions(n, factor_list)
ans = []
for (p, q) in partitions_of_n:
pIsPrime = numbthy.is_prime(p)
qIsPrime = numbthy.is_prime(q)
if pIsPrime and qIsPrime or (not pIsPrime and not qIsPrime):
continue
pseudo = (p - 1) * (q - 1)
lambda_n = carmichael_lambda_with_list(n, factor_list)
if pseudo % lambda_n == 0:
ans.append((p, q))
return ans
def fullyCompositeIdempotentPartitions(n, factor_list):
partitions_of_n = partitions(n, factor_list)
ans = []
for (p, q) in partitions_of_n:
if numbthy.is_prime(p):
continue
if numbthy.is_prime(q):
continue
pseudo = (p - 1) * (q - 1)
lambda_n = carmichael_lambda_with_list(n, factor_list)
if pseudo % lambda_n == 0:
ans.append((p, q))
return ans
def GF(n, poly=[], var='x', fmtspec="p"):
"""A shorthand for generating finite fields. If poly is not specified then one will be chosen from a list of Conway polynomials."""
if numbthy.is_prime(n):
if len(poly) < 2:
return FiniteField(n, [1], var=var, fmtspec=fmtspec)
else:
return FiniteField(
n, poly, var=var, fmtspec=fmtspec
) # Explicit characteristic and polynomial modulus
else: # n is not prime - hope it's a prime power
nfactors = numbthy.factor(n)
if (len(nfactors) != 1):
raise ValueError(
'GF({0}) only makes sense for {0} a prime power'.format(n))
p = nfactors[0][0]
e = nfactors[0][1] # Prime p, exponent e
if (len(poly) == 0):
try:
poly = readconway('CPimport.txt', p, e)[:-1] # Assume monic
except (IOError, ValueError):
print(" Look for non-Conway primitive polynomial")
poly = findprimpoly(p, e)
else:
if nfactors[0][1] != len(poly):
raise ValueError(
'Polynomial {0} does not have degree {1}'.format(
poly + [1], nfactors[0][1]))
return FiniteField(p, poly, var=var, fmtspec=fmtspec)
def problem2(N):
N = decimal.Decimal(N)
# Use this routine to find A value that fits for N
A = int(N.sqrt()) + 1
index = 1
maxdiff = 2**19 # A - sqrt(N) < 2 ** 19
while (index < maxdiff):
x2 = A * A - N
x = int(x2.sqrt())
if x * x == x2:
p, q = A - x, A + x
if (p * q == N):
if is_prime(p) == True and is_prime(q) == True:
return p
index += 1
A += 1
print "No solution"
return 0, 0
def is_prime_function():
print("Testing -- is_prime function --")
print("Using numpy.random generating {0} numbers".format(testnumber))
for n in range(testnumber):
a = rd.randint(minimum, maximum)
tf1 = nmtf.is_prime(tf.constant(a))
nm1 = nm.is_prime(a)
print("Test {0}------------------------------".format(n + 1))
print("nmtf.is_prime({0}) = {1}".format(a, tf1.eval()))
print(" nm.is_prime({0}) = {1}".format(a, nm1))
print("")
def GF(n,name='x',modulus=[]):
if numbthy.is_prime(n):
if len(modulus)<2:
print "Haven't coded prime field GF({0}) yet - sorry".format(n)
raise ValueError("Haven't coded prime field GF({0}) yet - sorry".format(n))
else:
return FiniteField(n,modulus,name) # Explicit characteristic and polynomial modulus
else: # n is not prime - hope it's a prime power
nfactors = numbthy.factor(n)
if (len(nfactors) != 1): raise ValueError('GF({0}) only makes sense for {0} a prime power'.format(n))
p = nfactors[0][0]; e = nfactors[0][1] # Prime p, exponent e
if (len(modulus) == 0):
try:
modulus = readconway('CPimport.txt',p,e)[:-1] # Assume monic
except(IOError,ValueError):
print(" Look for non-Conway primitive polynomial")
modulus = findprimpoly(p,e)
return FiniteField(p,modulus,name)
def GF(n, poly=[], var='x', fmtspec="p"):
"""A shorthand for generating finite fields. If poly is not specified then one will be chosen from a list of Conway polynomials."""
if numbthy.is_prime(n):
if len(poly)<2:
return FiniteField(n,[1],var=var,fmtspec=fmtspec)
else:
return FiniteField(n,poly,var=var,fmtspec=fmtspec) # Explicit characteristic and polynomial modulus
else: # n is not prime - hope it's a prime power
nfactors = numbthy.factor(n)
if (len(nfactors) != 1): raise ValueError('GF({0}) only makes sense for {0} a prime power'.format(n))
p = nfactors[0][0]; e = nfactors[0][1] # Prime p, exponent e
if (len(poly) == 0):
try:
poly = readconway('CPimport.txt',p,e)[:-1] # Assume monic
except(IOError,ValueError):
print(" Look for non-Conway primitive polynomial")
poly = findprimpoly(p,e)
else:
if nfactors[0][1] != len(poly):
raise ValueError('Polynomial {0} does not have degree {1}'.format(poly+[1],nfactors[0][1]))
return FiniteField(p,poly,var=var,fmtspec=fmtspec)
def GF(n, name='x', modulus=[]):
if numbthy.is_prime(n):
if len(modulus) < 2:
return FiniteField(n, [1], name)
else:
return FiniteField(
n, modulus,
name) # Explicit characteristic and polynomial modulus
else: # n is not prime - hope it's a prime power
nfactors = numbthy.factor(n)
if (len(nfactors) != 1):
raise ValueError(
'GF({0}) only makes sense for {0} a prime power'.format(n))
p = nfactors[0][0]
e = nfactors[0][1] # Prime p, exponent e
if (len(modulus) == 0):
try:
modulus = readconway('CPimport.txt', p, e)[:-1] # Assume monic
except (IOError, ValueError):
print(" Look for non-Conway primitive polynomial")
modulus = findprimpoly(p, e)
return FiniteField(p, modulus, name)
def test_is_prime(self):
for testcase in ((1, False), (0, False), (-2, True), (19, True),
(1236, False), (1237, True), (321197185, False)):
self.assertEqual(numbthy.is_prime(testcase[0]), testcase[1])