What problems have been frequently computationally verified for large values?

Question

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.

Answer 1

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.

Answer 2

Whether or not there exist any odd perfect numbers is an open problem. Numbers up to $10^{1500}$ have been checked (as of $2012$) without any success.

Answer 3

The Goldbach Conjecture has been verified up though $4\times 10^{18}$ by Oliviera e Silva (as of 2012). The history of these computations (13 previous records) can be found on Mathworld.

The Riemann Hypothesis has been verified through $10^{13}$ by X Gourdon (2004). The history of these computations can be found on Wikipedia.

The Union-closed Set Conjecture has been verified up to sets of size $46$ as well as for other special cases. The specific lower bound of size $46$ was found by Roberts and Simpson in 2010. The previous records were 18 (Sarvate and Renaud 1990) and 40 (Roberts 1992). Mathworld lists several other results that fail to beat Roberts 1992.

Answer 4

The Collatz conjecture, also called the $3x+1$ conjecture or Hail Stone sequence has been verified up to $87\times 2^{60}$ as of $2017$. More information can be found on Wikipedia.

Answer 5

Firoozbakht's conjecture states that, if $p_n$ is the $n$^th prime numbers, then the sequence $\left(\sqrt[n]{p_n}\right)_{n\in\mathbb N}$ is strictly decreasing. It has never been proved, but it it has already been checked for the primes below $10^{19}$.

Answer 6

• The Hadamard Conjecture states that a Hadamard matrix of order $4k$ should exist for every positive integer $k$. It has been numerically verified for all orders up to 668.
• The Circulant Hadamard Conjecture posits that there are no circulant Hadamard matrices of order $>4$, and has been verified numerically for most values up to $10^4$.

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If you allow liberal interpretation of "large values" as "high confidence", several of the Millennium Prize Problems provide examples.

• The mass gap part of the Yang-Mills Existence and Mass Gap problem has been numerically verified using lattice QCD. To do this, you discretize space on a lattice and evaluate the spectrum of the Hamiltonian, and refining the computation is done by using a finer lattice. At this point the numerical evidence is so overwhelming that it's not really meaningful to ask "to what bound has it been checked?"; we "know" that the mass gap part of this problem is true--the only challenge is proving it rigorously. In the past, though, this sort of computation was at the very frontier of supercomputing, and there was a time when verifying the existence of a mass gap (and related phenomena) numerically was a major industry.
• The origin of the Birch and Swinnerton-Dyer Conjecture was in number crunching on elliptic curves. The content of the conjecture is to rigorously prove certain trends which are observed numerically. I don't know how well those trends are now established numerically (but the linked Wikipedia article has plots showing roughly $10^6$ data points).
• The Riemann Hypothesis has been verified numerically to something like ten trillion zeroes.

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Answer 7

Searching of solutions of the Diophantine equation

$$x^3+y^3+z^3=k$$

for small $k$ ($k<1000$) has been performed for $|x|,|y|,|z|$ up to $10^{15}$.

For $k<100$, solutions were not yet found for $k=33$ and $k=42$. For $k<1000$, there are 14 values without solution.

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Edit: 33 has been cracked.

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Edit2: 42 has been cracked.