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

Author:
Don Coppersmith

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

MSC:
Primary 11D41

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]**Andrew Granville and Michael 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), no. 1, 329–359. MR**927694**, 10.1090/S0002-9947-1988-0927694-5**[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 Emma Lehmer,*On the first case of Fermat’s last theorem*, Bull. Amer. Math. Soc.**47**(1941), 139–142. MR**0003657**, 10.1090/S0002-9904-1941-07393-3**[4]**Barkley Rosser,*On the first case of Fermat’s last theorem*, Bull. Amer. Math. Soc.**45**(1939), 636–640. MR**0000025**, 10.1090/S0002-9904-1939-07058-4**[5]**Daniel Shanks and H. C. Williams,*Gunderson’s function in Fermat’s last theorem*, Math. Comp.**36**(1981), no. 153, 291–295. MR**595065**, 10.1090/S0025-5718-1981-0595065-7**[6]**Jonathan W. Tanner and Samuel S. Wagstaff Jr.,*New bound for the first case of Fermat’s last theorem*, Math. Comp.**53**(1989), no. 188, 743–750. MR**982371**, 10.1090/S0025-5718-1989-0982371-4

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

Article copyright:
© Copyright 1990
American Mathematical Society