<P> In fact, MF (p, r) = Φ (2), where Φ is the cyclotomic polynomial . </P> <P> The simplest generalized Mersenne primes are prime numbers of the form f (2), where f (x) is a low - degree polynomial with small integer coefficients . An example is 2 − 2 + 1, in this case, n = 32, and f (x) = x − x + 1; another example is 2 − 2 − 1, in this case, n = 64, and f (x) = x − x − 1 . </P> <P> It is also natural to try to generalize primes of the form 2 − 1 to primes of the form b − 1 (for b ≠ 2 and n> 1). However (see also theorems above), b − 1 is always divisible by b − 1, so unless the latter is a unit, the former is not a prime . There are two ways to deal with that: </P> <P> In the ring of integers (on real numbers), if b − 1 is a unit, then b is either 2 or 0 . But 2 − 1 are the usual Mersenne primes, and the formula 0 − 1 does not lead to anything interesting (since it is always − 1 for all n> 0). Thus, we can regard a ring of "integers" on complex numbers instead of real numbers, like Gaussian integers and Eisenstein integers . </P>

If n is prime then 2n −1 is also prime