Reminds me of that short story where aliens visit the Earth and a mathematician asks them if they proved Fermat's last theorem (presumably this was before Wile's proof) and the aliens replied, sure, we did. So the mathematician inquires about the proof and the aliens explained they tested it on all values up to some unimaginable number, leaving the mathematician deeply saddened. But I can't remember where I read it. Possibly Singh's book on FLT.
I made a tweet on our MHB Twitter account about the Polya Conjecture over a year ago. It's one of my favorite examples of why proofs are needed. It seems so unlikely to reverse direction until it does.