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 2008-12-15, 00:36 #1 Random Poster     Dec 2008 179 Posts Density of Mersenne divisors Suppose we have n consecutive primes starting with p; out of these, we would expect n*(some function of p) to divide a Mersenne number with prime exponent. What is known about this function?
2008-12-15, 01:14   #2
R.D. Silverman

"Bob Silverman"
Nov 2003
North of Boston

2·33·139 Posts

Quote:
 Originally Posted by Random Poster Suppose we have n consecutive primes starting with p; out of these, we would expect n*(some function of p) to divide a Mersenne number with prime exponent. What is known about this function?
A lot is known.

q = 2p+1 will divide M_p if q is a q.r. of p This happens 1/2 the time.
q = 2kp + 1 will divide M_p if q is a '2k'th power residue of p. When
and how frequently this happens is determined by Artin's higher reciprocity
laws.

It is an exercize for a grad student in analytic number theory to
estimate the function you seek.....

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