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# Fermat's Last Theorem (Case 1) and the Wieferich Criterion

Don Coppersmith
Mathematics of Computation
Vol. 54, No. 190 (Apr., 1990), pp. 895-902
DOI: 10.2307/2008518
Stable URL: http://www.jstor.org/stable/2008518
Page Count: 8
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## Abstract

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 $r \operatorname{mod} p^2$ such that $r^p \equiv r \operatorname{mod} p^2$, and thereby improve the lower bound on $p$ to $7.568 \times 10^{17}$.

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