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As an example, it is certainly trivial to verify that Fermat's Last Theorem is true for all numbers up to 10^10 using a reasonably powerful computer. We now have mathematical proof of that theorem for all numbers, but are there other examples showing the opposite? E.g. a certain theorem is known to be true for all numbers up to 10^20, but there is also a known counter example at a much larger value.
Obviously it's easy to make up such a theorem for the purpose of the question, so I would like to limit the scope to theorems that were actually researched by the mathematical community.