New bound for the first case of Fermat's last theorem

We present an improvement to Gunderson's function, which gives a lower bound for the exponent in a possible counterexample to the first case of Fermat's "Last Theorem," assuming that the generalized Wieferich criterion is valid for the first n prime bases. The new function increases beyond $n = 29$, unlike Gunderson's, and it increases more swiftly. Using the recent extension of the Wieferich criterion to $n = 24$ by Granville and Monagan, the first case of Fermat's "Last Theorem" is proved for all prime exponents below 156, 442, 236, 847, 241, 729.