# Are there theorems in number theory which are true for a large interval, but are known to be false for an arbitrarily large number outside that range? [duplicate]

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.

## marked as duplicate by M. Winter, Dietrich Burde, Community♦Oct 7 '17 at 11:12

• The Goodstein-sequences are even more mind-boggling! Even with start value $4$, the sequence grows so fast, that brute force calculation would never show that the sequence eventually reaches $0$. But we know that every such sequence terminates ending with $0$. Surprisingly, the peano axioms cannot prove that every Goodstein-sequence eventually reaches $0$, but in $ZFC$, there is a relatively easy proof. – Peter Oct 7 '17 at 17:24
Consider the difference between $\pi(x)$ and $\operatorname{li}(x)$, whose sign changes infinitely many times, as was proved by Littlewood.