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

Calculations are rarely involved in proving a mathematical theorem.

Let’s look at the example you used. The Collatz conjecture makes a claim about every positive integer. We aren’t going to be able to prove the Collatz conjecture by brute force, because we can’t check every positive integer (there are infinitely many).

If a proof is found for/against the Collatz conjecture, it will not be found by “checking” a ton of numbers, but instead will be found by logic and clever reasoning.