As an interesting example where we did have a finite bound that was still out of reach to computationally exhaust, see the Weak Goldbach Conjecture: https://en

As an interesting example where we did have a finite bound that was still out of reach to computationally exhaust, see the Weak Goldbach Conjecture: https://en.m.wikipedia.org/wiki/Goldbach%27s_weak_conjecture https://en.m.wikipedia.org/wiki/Goldbach%27s_weak_conjecture It was known to have at most finitely many counter-examples by the 1930’s, and by the 1950’s it was known that the largest counter-example had to be < 3^3^15. In 2002 that was lowered to about 10^1346. Still well out of reach of any computer! Only in 2012 was an unconditional proof given.