According to Wikipedia, the largest known prime is $2^{57,885,161}-1$ with $17,425,170$ digits.

Because a probable prime is usually easier to find than a proven prime (although for the Mersenne-primes, there is an algorithm to prove primilaty as fast as a probable prime test), I wonder if there is a larger known probable prime.

The same for twin primes, the largest known pair is $3,756,801,695,685\times 2^{666,669}\pm1$ with $200,700$ digits. Is there known a larger pair for which both entries are probable primes ?

  • 2
    $\begingroup$ Probably known but not published. Note that we are talking about ranges where being satisfied with a highly probable prime is not useful, and someone publishing that they found a probable prime is something I find unlikely. Then again, there may be a community out there where probable primes, being candidates for primeness-provers, are collected. And as you say, for "structured" primes such as Mersenne, a definite proof can be as fast as a probale test. $\endgroup$ – Hagen von Eitzen Sep 8 '15 at 21:34

The largest collection of (large) probable primes that I have seen is that of Henri & Renaud Lifchitz:


The largest PRP there is $(2^{13372531}+1)/3$ which is much smaller than $2^{57 885161}-1$ (about a quarter the number of digits). Generally, PRPs take as much effort to find as Mersenne primes of a similar size, but more effort is put toward Mersenne primes because of interest, convenient software, greater publicity, and the EFF prizes.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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