Will Edgington commented:
> Chris Nash writes:
>
> The smallest factor of 2^p-1, p a prime, is at least as big as
> 2p+1. All factors of a Mersenne number of prime exponent are of the
> form 2kp+1 - similarly for all 'new' factors of a composite
> exponent (ie that haven't appeared in any Mersenne number with an
> exponent that is a factor of the original one).
>
> Everything here is correct except that many "primitive" factors of
> composite exponent Mersennes are _not_ congruent to 1 mod twice the
> exponent. There are numerous counter-examples, starting with 5, which
> is a factor of M(4) = 15 but not a factor of any smaller Mersenne.
> All "primitive" factors are, however, congruent to 1 mod the exponent,
> even for composite exponents and factors.
Both of Chris Nash's remarks are intended for odd p.
The factor 3 of 2^2 - 1 is not congruent to 1 modulo 2*2.
"Primitive factors" should be restricted to primitive prime factors.
The factor 1057 = 7*151 of 2^15-1 = 7*31*151 does not divide any
M(n) with 1 <= n < 15, yet 1057 is not 1 modulo 15
(however the new prime 151 has this form).
Peter Montgomery
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