It seems to Feynman's argument works via the opposite of ruling out coincidences of that sort. It shows "If nothing surprising happens, then we expect very near

It seems to Feynman's argument works via the opposite of ruling out coincidences of that sort. It shows "If nothing surprising happens, then we expect very nearly zero counterexamples to Fermat's Last Theorem"... But the "surprising" in "If nothing surprising happens" encompasses both surprising structure (things which are true for some deep, clean reason) AND surprising numeric coincidences (things which are true for no good reason, but just happen, surprisingly, to line up in a nice, unexpected way).