> Elsewhere, there is the Lenstra–Pomerance–Wagstaff > conjecture, suggesting that the number of Mersenne primes > with exponent p less than x is asymptotically approximated by > > e^y x log_2(x) where y is the Euler-Mascheroni constant. > > The mainstream view over the years in this mailing list seems > to have been that there is an infinite number of MPs, and > those (like me) who think otherwise have sometimes been > treated like flat-earthers. So has there been a change of > mood, or is Chris saying simply that a *proof* is a long way > away? - Yes, I know that mathematics is all about proofs.
Indeed that is what I am saying. You are correct the widely believed conjecture is that there are an infinite number of Mersennes. My exposition of this is at http://primes.utm.edu/mersenne/heuristic.html And yes, a proof is far away. Far far away. Your being a doubter is fine with me. Sometimes our "obvious" conjectures turn out to later contradict each other! Chris. ps: It took over 100 years to get from lim sup (p_{n+1}-p_n)/log p_n <= 1 to lim sup (p_{n+1}-p_n)/log p_n = 0 and if you accept RH(\epsilon): lim sup p_{n+1}-p_n <= 16. [This may now be 8, not sure off the top of my head.) Unconditionally lim sup (p_{n+1}-p_n) / [(log p_n)^(1/2) log log p_n)^2] < infinity To get a proof that there are infinitely many Mersennes will take major unprecedented advancements. _______________________________________________ Prime mailing list [email protected] http://hogranch.com/mailman/listinfo/prime
