def strong_pseudoprimes(a):
"""
Strong Pseudoprimes to Base a
Args:
a -- integer greater than 1
OEIS
"""
require_integers(["a"], [a])
require_geq(["a"], [a], 2)
for c in composites():
if c % 2 == 1:
d = c - 1
r = 0
d, r = factor_out_twos(d)
if (a**d) % c == 1:
yield c
continue
for e in range(0, r):
if pow(a, (d * 2**e), c) == c - 1:
yield c
continue
def lucas_pseudoprimes(P, Q):
"""
Lucas Pseudoprimes: Composite number passing the Lucas primality test for P and Q
Args:
P -- integer greater than zero
Q -- integer
OEIS
"""
require_integers(["P", "Q"], [P, Q])
require_geq(["P"], [P], 1)
D = P * P - 4 * Q
#rather than recompute we will step the sequence as needed
lucas_pos = 0
a, b = 0, 1
for c in composites():
if c % 2 == 1:
if gcd(c, Q) == 1:
delta = c - jacobi_symbol(D, c)
if delta > lucas_pos:
for i in range(delta - lucas_pos):
a, b = b, P * b - Q * a
lucas_pos += delta - lucas_pos
if a % c == 0:
yield c
def euler_jacobi_pseudoprimes():
"""
Euler-Jacobi Pseudoprimes\n
OEIS A047713
"""
for c in composites():
if c % 2 == 1:
if 2**((c - 1) // 2) % c == jacobi_symbol(2, c) % c:
yield c
def euler_pseudoprimes():
"""
Euler Pseudoprimes\n
OEIS A006970
"""
for c in composites():
if c % 2 == 1:
if 2**((c - 1) // 2) % c in (1, c - 1):
yield c
def strong_lucas_pseudoprimes(P, Q):
"""
Lucas Pseudoprimes: Composite numbers passing either of two the Lucas primality tests for P and Q\n
OEIS
"""
def lucas_U_test(n, d, P, Q):
a, b = 0, 1
for i in range(d):
a, b = b, (P * b - Q * a) % n
if a == 0:
return True
return False
def lucas_V_test(n, d, s, P, Q):
a, b = 2, P
ctr = d
nctr = 0
for k in range(ctr):
a, b = b, (P * b - Q * a) % n
nctr += 1
if a == 0:
return True
while ctr < d * (2**(s - 1)):
for k in range(ctr):
a, b = b, (P * b - Q * a) % n
nctr += 1
ctr *= 2
if a == 0:
return True
return False
D = P * P - 4 * Q
for n in composites():
if n % 2 == 0:
continue
if gcd(n, D) != 1:
continue
delta = n - jacobi_symbol(D, n)
d, s = factor_out_twos(delta)
# Check with a Lucas V-Sequence
if lucas_U_test(n, d, P, Q):
yield n
continue
# Check with a Lucas V-Sequence
if lucas_V_test(n, d, s, P, Q):
yield n
def carmichael_numbers():
"""
Charmichael Numbers: Composite numbers that are Fermat Pseudoprimes to all bases\n
OEIS A002997
"""
def all_fermat_test(n):
B = coprime_to(n)
for b in B:
if pow(b, n - 1, n) != 1:
return False
return True
for c in composites():
if all_fermat_test(c):
yield c
def fermat_pseudoprimes(a):
"""
Fermat Pseudoprimes to Base a
Args:
a -- integer greater than 1
OEIS A001567, A005935-A005939
"""
require_integers(["a"], [a])
require_geq(["a"], [a], 2)
for c in composites():
if pow(a, c - 1, c) == 1:
yield c
def weak_pseudoprimes(a):
"""
Weak Pseudoprimes to Base a
Args:
a -- integer greater than 1
OEIS
"""
require_integers(["a"], [a])
require_geq(["a"], [a], 2)
yield 1
for c in composites():
if pow(a, c, c) == a:
yield c
def pell_pseudoprimes_2():
"""
Pell Pseudoprimes: Passes a version of the Pell primality test\n
OEIS A099011
"""
lucas_pos = 0
a, b = 0, 1
for c in composites():
if c % 2 == 1:
for i in range(c - lucas_pos):
a, b = b, 2 * b + a
lucas_pos += c - lucas_pos
if (a - jacobi_symbol(2, c)) % c == 0:
yield c
def fibonacci_pseudoprimes():
"""
Fibonacci Pseudoprimes: Special case of Lucas Pseudoprimes, also excluding multiples of 5\n
OEIS A081264
"""
lucas_pos = 0
a, b = 0, 1
for c in composites():
if c % 2 == 1 and c % 5 != 0:
delta = c - jacobi_symbol(5, c)
if delta > lucas_pos:
for i in range(delta - lucas_pos):
a, b = b, b + a
lucas_pos += delta - lucas_pos
if a % c == 0:
yield c