How do mathematicians brute force large number sets

Question

How do mathematicians calculate extremely large sets of numbers to prove theorems? Both now and in the past?

A modern example is the Collatz Conjecture, 2\^68 numbers were checked, so what resources do mathematicians use for this? Supercomputers?

And in the past, the Polya Conjecture was disproven by C. Brian Haselgrove in 1956, he calculated more than 2\^361 values, so how was this done when computers weren’t as powerful?

In: 40

Answer 1

>And in the past, the Polya Conjecture was disproven by C. Brian Haselgrove in 1956, he calculated more than 2^361 values, so how was this done when computers weren’t as powerful?

Haselgrove didn’t actually do that calculation – he proved that there was an answer somewhere under 2×10^361, but he didn’t actually find the answer, he just proved that there was one.

Someone else did find one in 1960, at a MUCH lower level than that – 906,180,359. That’s just over 9×10^8 – you could multiply it by the number of atoms in the universe four times and it would STILL be lower than what Haselgrove proved.

But even they didn’t check every number – it wasn’t until 1980 that it was proven that the lowest counterexample was 906,150,257.