Twin Mersenne exponent conjecture

Question

Conjecture: (17, 19) is the largest twin Mersenne exponent pair and the only isolated twin Mersenne exponent pair. ((3, 5), (5, 7) are not isolated because they share 5.) Is there anybody who can prove it?

Thanks,

Liang

Answer 1

As Wojowu notes in their comment, given the state of the art, we can't even prove that there are infinitely many primes $p$ where $M_p$ is composite. That said, standard heuristic arguments support that there should be only finitely many twin prime pairs of this sort. We will show a heuristic here which shows that we should believe stronger claim: Namely there should be only finitely many pairs of consecutive primes $p$ and $q$ where $M_p$ and $M_q$ are both prime.

Heuristically, the Prime Number Theorem tells that the "chance" a number x is prime should be roughly $\frac{1}{\log x}$. Thus, given a prime $p$, the chance that $2^p-1$ is prime should be roughly $$\frac{1}{\log (2^p-1)} \approx \frac{1}{\log 2^p} \approx \frac{1}{p\log 2}. $$ Now, if $q$ is the prime after $p$, then $q \geq p+2$, so by the same logic the chance that $M_q$ is prime is no more than about $$\frac{1}{(p+2)\log 2} \approx \frac{1}{p\log 2}.$$

So the expected number of such prime pairs is bounded above by about $$\sum_{p \,\, \mathrm{prime}} \left(\frac{1}{p\log 2}\right)^2 \leq \left(\frac{1}{\log 2}\right)^2\sum \frac{1}{n^2}.$$

The series on the right-hand side obviously converges so we should expect only finitely many of these Mersenne pairs.

As always, some disclaimers exist here. We haven't proven anything here. The heuristic also assumes that there's no relationship between whether $M_p$ is prime and the chance that $M_q$ is prime. It would be nice to have some empirical observation on whether that appears to be the case, but given the very small number of Mersenne primes, such a pattern would be difficult to detect just from the data. The obvious source of that sort of problem is that one having a factor would cause another to not have a chance to have that factor. But if there were any substantial hidden dependence of that sort, one would expect it to go in the direction of making slightly more values of $M_q$ composite when $M_p$ is prime. So it shouldn't alter the direction of the heuristic.

Also, note that the series in question is converging quickly, enough that we could take a sum over all $n$ rather than just primes and still get convergence. So even some dependence among terms would be unlikely to be enough to get divergence. Since this series was the heuristic for the finiteness of a much denser set of candidates than your original question, we can be pretty confident that your original set of pairs is finite.

Given how quickly the series converges, it seems like your guess about (17,19) is likely to be correct that they are the largest such pair. However, we're absolutely nowhere near proving any result like this. Even proving there are infinitely many Mersenne composites would be a tremendous breakthrough.