Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I asked earlier whether it can be proved that infinitely many elements of $P_n$ for some positive value of $n$ (here $P_n$ refers to the set of numbers with at most $n$ prime divisors). There I received a positive answer in the form of a 1978 paper of Iwaniec which confirmed that suitable quadratic polynomials do in fact hit $P_2$ infinitely often.

My question now concerns the situation with Mersenne primes. What is the least value of $n$, if it is known to be finite, such that it is known that infinitely many Mersenne numbers $2^p - 1$ (where $p$ is a prime) hits $P_n$ infinitely often. In other words, there exist infinitely many primes $p$ such that $2^p - 1$ has at most $n$ prime factors.

Edit: in view of the relatively clear answer given by http://www.math.ucsd.edu/~asalehig/SG_AffineSieveExpandersOverviewMSRI.pdf, it seems that what I asked is an open problem without much chance of being resolved in the near future, so I ask this modified question:

Let $M(x) = \{2^p - 1 : p \leq x\}$ where $p$ as usual denotes a prime. Is it possible to find two functions $f(x), g(x)$ that tend to infinity with $f$ taking on values in the positive integers such that $$\displaystyle |M(x) \cap P_{f(x))}| \gg g(x)?$$

In other words, can it be shown that the set of Mersenne numbers with $p \leq x$ having relatively few prime factors (here $f$ is understood to tend to infinity slowly) tends to infinity (perhaps slowly) as $x$ tends to infinity?

share|improve this question
1  
Not quite relevant, but are you aware that it has never been proved that there are infinitely many Mersenne composites? –  Gerry Myerson Dec 8 '13 at 3:36
    
A similar question was raised on MO by Igor Rivin earlier: see mathoverflow.net/questions/81729/mersenne-quasi-primes –  Lucia Dec 8 '13 at 15:53

1 Answer 1

up vote 5 down vote accepted

This is a classical problem, and it remains open to show that there exist arbitrarily large Mersenne numbers with a bounded number of prime factors. The work of Bourgain, Gamburd, and Sarnak on the affine sieve may be seen as a generalization of this kind of question, but their work does not say anything for Mersenne numbers. An exposition of this work may be found in Kowalski's Bourbaki article http://arxiv.org/abs/1012.2793 ; also see these lecture notes of Golsefidy http://www.math.ucsd.edu/~asalehig/SG_AffineSieveExpandersOverviewMSRI.pdf which explicitly mentiones the connection with Mersenne numbers.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.