So the thing about conjectures for all positive integers is that proving them true is a horrendous idea by brute force—there’s an infinite number of positive integers. But wait! What if we invented a computer that could check the first n positive integers in 1 second, the next n positive integers in 1/2 second (I guess something is derivable about the next set from the previous set by some mechanical analogue of strong induction or something?), the next n positive integers in 1/4 second, the next n positive integers in 1/8 second, etc., we’d be able to either prove the conjecture true by exhaustion or find a counterexample in 2 seconds. There is of course a huge realistic problem: we’ll probably find an easier way around this general issue sooner than we invent infinite computer speed or conquer the Planck Time. Thus, this question deals with idealistic usage of computers, which is quite an interesting combination.