Timeline for Mersenne Prime Sequences
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| May 13, 2016 at 0:22 | comment | added | reuns | @JoeSilverman : yes I know that, but it is independent of the fact that $2^{2}-1$,$2^{2^2-1}-1$,$2^{2^{2^2-1}-1}-1,2^{2^{2^{2^2-1}-1}-1}-1$ are all primes (even if only the two last really matter). the only fact I know is that $2^{2^{31}-1}-1$ has been shown composite, but it is a different double Mersenne iterated sequence. but if you look on wiki/Double_Mersenne_number you'll think that the empirical probability that $2^{2^p-1}-1$ is prime when $2^p-1$ is prime is quite.... high. | |
| May 13, 2016 at 0:16 | comment | added | Joe Silverman | @user1952009 Let $a_n$ be a sequence of positive integers whose entries don't have any particular reason to be composite, for example the sequence you suggest $2^{2^n}-1$. The "probability" that $a_n$ is prime is roughly $1/\log a_n$, so a rough heuristic is that if $\sum_{n=1}^\infty \frac{1}{\log a_n}$ converges, then the sequence contains finitely many primes, and if it diverges, then the sequence contains infinitely many primes. N.B. This is just a very coarse heuristic, it generally needs to be refined to take account of congruence conditions. But since you ask for a reason, there is it. | |
| May 13, 2016 at 0:08 | comment | added | reuns | @JoeSilverman : but here it is an exponential iterated sequence, is there any reason to think that $\displaystyle 2^{\textstyle 2^{127}-1}-1$ is not prime and that there are not an infinity of primes when continuing the sequence ? | |
| Jun 25, 2013 at 3:02 | review | First posts | |||
| Jun 26, 2013 at 0:08 | |||||
| Jun 17, 2013 at 18:35 | comment | added | Joe Silverman | I think that alias is overstating what's been proven. There are certain elliptic divisibility sequences and certain polynomial orbits for which one can prove there are only finitely many primes. Some of these, in particular the ones that Graham Everest and his colleagues/students studied, can be somewhat subtle to prove; but roughly speaking, they all come from cases where there is a "map" from some other sequence that provides a nontrivial divisor. Of course, there are also trivial examples. But in general, it is only conjectured that EDS and poly orbits have finitely many primes. | |
| Jun 17, 2013 at 16:53 | comment | added | Gerhard Paseman | Euclid proved the opposite, for a certain iterated sequence, and thus for infinitely many other iterated sequences. Later Dirichlet improved upon this by showing it held for every possible permissible instance of the iterated sequence. Gerhard "Is Talking About Iterated Adding" Paseman, 2013.06.17 | |
| Jun 17, 2013 at 16:34 | comment | added | user9072 | Welcome to MO! Could you perhaps provide some more details. (You can expand the answer via clicking 'edit' just below it.) | |
| Jun 17, 2013 at 16:30 | history | answered | alias | CC BY-SA 3.0 |