Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether the set of Mersenne primes is finite or infinite. The Lenstra–Pomerance–Wagstaff conjecture asserts that there are infinitely many Mersenne primes and predicts their order of growth. It is also not known whether infinitely many Mersenne numbers with prime exponents are composite, although this would follow from widely believed conjectures about prime numbers, for example, the infinitude of Sophie Germain primes congruent to 3 (mod 4). For these primes , (which is also prime) will divide , for example, , , , , , , , and . Since for these primes , is congruent to 7 mod 8, so 2 is a quadratic residue mod , and the multiplicative order of 2 mod must divide formula_1 = . Since is a prime, it must be or 1. However, it cannot be 1 since formula_2 and 1 has no prime factors, so it must be . Hence, divides formula_3 and formula_4 cannot be prime.