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There are several conjectures in Mathematics that seem to be true but have not been proved. Of course, as computing power increased, folks have expanded their search for counterexamples ever and ever upwards.
Providing a counterexample to a conjecture with a very large number would be interesting, but I cannot think of any non-trivial examples where a really large number has been found to disprove a (non-trivial) conjecture. I've seen plenty of large numbers serving as bounds to some value, but usually this is something known to be bounded (i.e. finite) anyway.
Out of curiosity, what's the largest counterexample you've seen to disprove a conjecture?