Fermat's last theorem (case ) and the Wieferich criterion

Author:
Don Coppersmith

Journal:
Math. Comp. **54** (1990), 895-902

MSC:
Primary 11D41

DOI:
https://doi.org/10.1090/S0025-5718-1990-1010598-2

MathSciNet review:
1010598

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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 such that , and thereby improve the lower bound on *p* to .

**[1]**A. Granville and M. B. Monagan,*The first case of Fermat's last theorem is true for all prime exponents up to*714,591,416,091,389, Trans. Amer. Math. Soc.**306**(1988), 329-359. MR**927694 (89g:11025)****[2]**N. G. Gunderson,*Derivation of criteria for the first case of Fermat's last theorem and the combination of these criteria to produce a new lower bound for the exponent*, Thesis, Cornell University, 1948.**[3]**D. H. Lehmer and E. Lehmer,*On the first case of Fermat's last theorem*, Bull. Amer. Math. Soc.**47**(1941), 139-142. MR**0003657 (2:250f)****[4]**B. Rosser,*On the first case of Fermat's last theorem*, Bull. Amer. Math. Soc.**45**(1939), 636-640. MR**0000025 (1:5b)****[5]**D. Shanks and H. C. Williams,*Gunderson's function in Fermat's last theorem*, Math. Comp.**36**(1981), 291-295. MR**595065 (82g:10004)****[6]**J. W. Tanner and S. S. Wagstaff, Jr.,*New bound for the first case of Fermat's Last Theorem*, Math. Comp.**53**(1989), 743-750. MR**982371 (90h:11028)**

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DOI:
https://doi.org/10.1090/S0025-5718-1990-1010598-2

Article copyright:
© Copyright 1990
American Mathematical Society