I just joined GIMPS (now 6% done testing a number with exponent just
short of 10M if it makes a difference) and I have been looking into the
theory behind Mersenne primes.
Can anyone show me or at least point me to a webpage with the proof that
the exponent of a Mersenne prime must be prime? How about a proof that the
LL test works? I have had math through DiffEq I. It is intuitively obvious
to me that every Mersenne number with even composite exponent will be found
by the formula M(p) = 4M(p-2)+3.
Since M(2)=3 is a multiple of 3, all these numbers will also be multiples,
and therefore composite. However, I can't understand why this is true of
numbers whose exponents have higher
This is particularly easy to see when the numbers are written in binary.
Since it is difficult to hand-compute the factors of Mersenne composites
much above 1023, I cannot easily search for patterns in higher Mersenne
numbers with composite exponent that has no factor of 2.
A completely unrelated question: Why, on the PrimeNet stats summery page,
are far more numbers listed as "finished LL" than as "available for
doublecheck"? Does this simply mean that some numbers do not require
doublechecking because they were turned in by a proven computer? Or have
they been already double-checked and turned in since the page was last
updated?
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