Primes Generated by Recurrence Sequences

will contain infinitely many prime terms — known as Mersenne primes. The Mersenne prime conjecture is related to a classical problem in number theory concerning perfect numbers. A whole number is said to be perfect if, like 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14, it is equal to the sum of all its divisors apart from itself. Euclid pointed out that 2k−1(2k − 1) is perfect whenever 2 − 1 is prime. A much less obvious result, due to Euler, is a partial converse: if n is an even perfect number, then it must have the form 2k−1(2k−1) for some k with the property that 2 − 1 is a prime. Whether there are any odd perfect numbers remains an open question. Thus finding Mersenne primes amounts to finding (even) perfect numbers. The sequence M certainly produces some primes initially, for example