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.
Will
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