def _get_factorized(base, pow):
x = _integers.get((base, pow), None)
if x is None:
x = Integer(base**pow - 1)
x.factorize()
_integers[(base, pow)] = x
return x
def _get_factorized(base, pow):
x = _integers.get((base, pow), None)
if x is None:
x = Integer(base ** pow - 1)
x.factorize()
_integers[(base, pow)] = x
return x
def _min_power(p, n, primes):
"""Minimal power t such that \pi(p(p^t-1)(p^2t-1)...(p^nt-1)) contains given set
of primes.
"""
t = 0
primes = set(primes)
primes.remove(p)
while primes:
t += 1
x = Integer()
for i in range(1, n + 1):
x *= _get_factorized(p, t * i)
#x = prod(Integer(p ** (t * i) - 1) for i in range(1, n + 1))
x.factorize()
primes -= set(x.factors)
return t
def _min_power(p, n, primes):
"""Minimal power t such that \pi(p(p^t-1)(p^2t-1)...(p^nt-1)) contains given set
of primes.
"""
t = 0
primes = set(primes)
primes.remove(p)
while primes:
t += 1
x = Integer()
for i in range(1, n + 1):
x *= _get_factorized(p, t * i)
# x = prod(Integer(p ** (t * i) - 1) for i in range(1, n + 1))
x.factorize()
primes -= set(x.factors)
return t