20081215, 00:36  #1 
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?

20081215, 01:14  #2  
"Bob Silverman"
Nov 2003
North of Boston
2·3^{3}·139 Posts 
Quote:
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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