What Is The Largest Non Proth And Non Marsenne Prime?

[ToraToraTora]

Google is a right pain in the April with this sort of search. the powers of 2 +1 and -1 are a bit of a short cut with prime identification. i wonder how many have been omitted using this shortcut. Are there undiscovered primes lower than out largest known?

[Buenchico]

Relevant discussion:
https://www.reddit.com/r/math/comments/21nclc/do_we_know_every_prime_smaller_than_the_largest/

See also here:
https://googology.fandom.com/wiki/Largest_known_prime

[jim360]

There are a few other classes that lead to large prime numbers. The generalised Fermat primes, of the form a^(2^k) + b^(2^k), are one such class. Typically in (large) prime searches, one takes b = 1, and you can see that there's still a power of two floating around.

Also, there's this one: define Phi_3(x) = 1+ x + x^2 , then for certain values of x, this is prime. Again, the relatively compact form allows this to be looked at nicely. The largest example has x of the form

x = -a^(2^19) , a = 123447 = 3*41149

and the next-largest is

x = - b^(3*2^17), b = 143332 = 2^2*7*5119

Realistically, it's hard to move too far away from these sorts of expression. Mersenne numbers as well have comparatively "easy" tests for primality, meaning that it's not nearly so hard to code up a search for them. In particular, you can determine if a number 2^p-1 is composite without having to factor it. Picking a random large number with no structure, however, is much harder to test for primality, meaning that you might have to just search for factors, which is (currently) extremely difficult; while quantum algorithms may be faster in the future, they are some way away from being applicable to searches for large primes.