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If I investigate the Goldbach conjecture by testing individual even integers to verify that they accord with it, do I have more reason to believe that the conjecture is true the more integers I verify? Or am I in just the same epistemic position regarding the conjecture whether I've verified one integer or a billion?

### As you clearly know, no

As you clearly know, no matter how many integers you have checked, that will always be a finite set, and so there will always be infinitely many integers you have not checked. Unless you had some reason to believe that a counterexample to Goldbach must be "low", then, it's hard to see why your checking a handful of cases should give you any more confidence that Goldbach is true. But there are some weird issues about how probability behaves in such cases, about which Timothy WIlliamson and others have written.

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