Although any theorem (or true conjecture) can be computationally checked, many long-standing open problems have been computational verified for very large values. For example, the Collatz Conjecture and Fermat's Last Theorem (before it was proven) were computationally verified by large scale computation programs. Not only have these calculations been carried out, but there is a lengthy history of improving the bound for which these calculations have been carried out until.
What are other problems (not necessarily from number theory) have been similarly verified for values up to some large bound, and how high have they been checked? Specifically I’m interested in cases where is an established history of computationally verifying the problem up to larger and larger bounds.
I’m interested both in the current cutting edge and the history of the computation.
Best Answer
The nonexistence of Fermat primes beyond $65537$ has been verified, as far as I know, to $2^{2^{32}}+1$. Thereby, we have identified the last constructible regular prime-sided polygon up to a point far beyond where any such construction could be carried out. (Based on current theories of quantum gravity, a regular polygon having the shortest possible side length and "only" $2^{2^8}+1$ sides would not fit in the known Universe.)
It was, of course, Euler who first killed Fermat's conjecture that $2^{2^n}+1$ is prime for all natural numbers $n$, by disproving it for $n=5$. Now the opposite conjecture is in vogue, and it has been verified up to $n=32$. Testing of Fermat numbers for primality can be accomplished by Pepin's Test, a stronger form of Fermat's Little Theorem whereby a Fermat number $M\ge 5$ is prime iff $3^{(M-1)/2}\equiv -1 \bmod M$. Because Pepin's test does not directly identify factors when the number is composite, $2^{2^n}+1$ has no known factors, despite being certified composite, for $n=20$ and $n=24$.
See here for a more thorough discussion of Fermat numbers.