Fermat’s last theorem (case $1$) and the Wieferich criterion
HTML articles powered by AMS MathViewer
- by Don Coppersmith PDF
- Math. Comp. 54 (1990), 895-902 Request permission
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 $r\bmod {p^2}$ such that ${r^p} \equiv r\bmod {p^2}$, and thereby improve the lower bound on p to $7.568 \times {10^{17}}$.References
- 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, DOI 10.1090/S0002-9947-1988-0927694-5 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.
- D. H. Lehmer and Emma Lehmer, On the first case of Fermat’s last theorem, Bull. Amer. Math. Soc. 47 (1941), 139–142. MR 3657, DOI 10.1090/S0002-9904-1941-07393-3
- Barkley Rosser, On the first case of Fermat’s last theorem, Bull. Amer. Math. Soc. 45 (1939), 636–640. MR 25, DOI 10.1090/S0002-9904-1939-07058-4
- Daniel Shanks and H. C. Williams, Gunderson’s function in Fermat’s last theorem, Math. Comp. 36 (1981), no. 153, 291–295. MR 595065, DOI 10.1090/S0025-5718-1981-0595065-7
- 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, DOI 10.1090/S0025-5718-1989-0982371-4
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Math. Comp. 54 (1990), 895-902
- MSC: Primary 11D41
- DOI: https://doi.org/10.1090/S0025-5718-1990-1010598-2
- MathSciNet review: 1010598