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

Wouldn’t a proof by induction take away the need for a brute force approach?