If Riemann was proved, would Wagstaff be proved?

eepiccolo · 2003-07-30, 16:03

I was looking at the prime pages to try to figure this out, but I wanted conformation from someone more knowledgeable in mathematics :) .

If the Riemann Hypothesis was proven, would the Wagstaff conjecture become a proof? That is to say, would it be proved that log2( log2( nth Mersenne Prime)) vs. n would grow approximately linearly with slope 2^(1/e^gamma)? I think the proof the there are infinite Mersenne primes falls out of this, also.

ewmayer · 2003-07-30, 17:31

The Wagstaff Conjecture is a heuristic, probabilistic argument about the expected distribution of Mersenne primes, based on the known form of factors of prime-exponent Mersenne numbers and some basic number theory. As such, it may not be something provable, like a theorem. Even if it were somehow provable that yes, Mersenne primes should follow the expected Poisson distribution, I'm not sure that would be quite the same as proving that there are infinitely many - it would make it overwhelmingly likely that there are, but in a statistical sense. Perhaps someone with a background in statistics would care to comment on this.

I'm not sure whether the Riemann Hypothesis is involved in any crucial way in the conjecture - a cornerstone of the heuristics is Mertens' Theorem, which involves the Riemann zeta function, but is in fact a proven theorem - it does not require GRH to be true:

http://mathworld.wolfram.com/MertensTheorem.html

I'd like to dig a bit deeper, but am unable to access the Prime Pages just now - I keep getting "connection refused" errors.

2003-07-30, 16:03	#1
eepiccolo Dec 2002 Frederick County, MD 172₁₆ Posts	If Riemann was proved, would Wagstaff be proved? I was looking at the prime pages to try to figure this out, but I wanted conformation from someone more knowledgeable in mathematics :) . If the Riemann Hypothesis was proven, would the Wagstaff conjecture become a proof? That is to say, would it be proved that log2( log2( nth Mersenne Prime)) vs. n would grow approximately linearly with slope 2^(1/e^gamma)? I think the proof the there are infinite Mersenne primes falls out of this, also.

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2003-07-30, 17:31	#2
ewmayer ∂²ω=0 Sep 2002 República de California 11639₁₀ Posts	The Wagstaff Conjecture is a heuristic, probabilistic argument about the expected distribution of Mersenne primes, based on the known form of factors of prime-exponent Mersenne numbers and some basic number theory. As such, it may not be something provable, like a theorem. Even if it were somehow provable that yes, Mersenne primes should follow the expected Poisson distribution, I'm not sure that would be quite the same as proving that there are infinitely many - it would make it overwhelmingly likely that there are, but in a statistical sense. Perhaps someone with a background in statistics would care to comment on this. I'm not sure whether the Riemann Hypothesis is involved in any crucial way in the conjecture - a cornerstone of the heuristics is Mertens' Theorem, which involves the Riemann zeta function, but is in fact a proven theorem - it does not require GRH to be true: http://mathworld.wolfram.com/MertensTheorem.html I'd like to dig a bit deeper, but am unable to access the Prime Pages just now - I keep getting "connection refused" errors.