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# -*- coding: utf-8 -*-
"""
In mathematics, the Lucas–Lehmer test (LLT) is a primality test for Mersenne numbers.
https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test
A Mersenne number is a number that is one less than a power of two.
That is M_p = 2^p - 1
https://en.wikipedia.org/wiki/Mersenne_prime
The Lucas–Lehmer test is the primality test used by the
Great Internet Mersenne Prime Search (GIMPS) to locate large primes.
"""
# Primality test 2^p - 1
# Return true if 2^p - 1 is prime
def lucas_lehmer_test(p: int) -> bool:
"""
>>> lucas_lehmer_test(p=7)
True
>>> lucas_lehmer_test(p=11)
False
# M_11 = 2^11 - 1 = 2047 = 23 * 89
"""
if p < 2:
raise ValueError("p should not be less than 2!")
elif p == 2:
return True
s = 4
M = (1 << p) - 1
for i in range(p - 2):
s = ((s * s) - 2) % M
return s == 0
if __name__ == "__main__":
print(lucas_lehmer_test(7))
print(lucas_lehmer_test(11))