My understanding is that for any general formula to be accepted, there has to be a proof for all 'n', by construction or by negation.
Even if the formula can be tested for high values of 'n', we merely say it works up to the tested limit but not beyond.
Do we have some non trivial examples where a formula works till a certain high 'n', say n = 100 or 1000, but not beyond that?
I'm curious because we often have unproven formulae that work correctly for an arbitrary 'n', which one may test for; and neither counter examples are found, nor a proof. But I don't see famous cases where a counter example could disprove a 'promising' formula.
Yes,you may find many formulae which are true up to certain n and not true for other for integers larger than n. For example the statement $$(n-1)(n-2)(n-3)=0$$ is true for $n=1,2,3$ but is false for $ n>3.$