The Fermat number $F_6$ was shown to have nontrivial factorization, by Landry at the age of 82. And apparently it was Landry's only mathematical publication.

(Source: Ribenboim, Prime number records(the smaller book).)

This is perhaps not a "major mathematical advance" in the sense of Hardy; but is inspiring nonetheless. I have seen a good number of elderly retired people with dreams of solving Fermat's Last Theorem or other such theorems in a simple way, and doggedly keep on trying and without getting disheartened by the lack of recognition for their efforts.