Fermat’s last theorem (Case 1) and the wieferich criterion

This note continues work by the Lehmers [3], Gunderson [2], Granville and Monagan [1], and Tanner and Wagstaff [6], producing lower bounds for the prime exponent p in any counterexample to the first case of Fermat’s Last Theorem. We improve the estimate of the number of residues r mod p2such that rP= r mod p2and thereby improve the lower bound on p to 7.568 x 1017. © 1990 American Mathematical Society.