The continuing search for Wieferich primes

Knauer, Joshua; Richstein, Jörg

doi:10.1090/S0025-5718-05-01723-0

The continuing search for Wieferich primes
HTML articles powered by AMS MathViewer

by Joshua Knauer and Jörg Richstein PDF

Math. Comp. 74 (2005), 1559-1563 Request permission

Abstract:

A prime $p$ satisfying the congruence \[ 2^{p-1} \equiv 1 \pmod {p^2}\] is called a Wieferich prime. Although the number of Wieferich primes is believed to be infinite, the only ones that have been discovered so far are $1093$ and $3511$. This paper describes a search for further solutions. The search was conducted via a large scale Internet based computation. The result that there are no new Wieferich primes less than $1.25 \cdot 10^{15}$ is reported.

References

Abel, N. H. (1828). Journal für die reine und angewandte Mathematik, 3:212.
Beeger, N. (1914). Quelques remarques sur les congruences $r^{p-1} \equiv 1 \pmod {p^2}$ et $(p-1)! \equiv -1 \pmod {p^2}$. Messenger of Mathematics, 43:72–84.
Beeger, N. (1922). On a new case of the congruence $2^{p-1} \equiv 1 \pmod {p^2}$. Messenger of Mathematics, 51:149–150.
N. G. W. H. Beeger, On the congruence $2^{p-1}\equiv 1\pmod p^2$ and Fermat’s last theorem, Nieuw Arch. Wiskde 20 (1939), 51–54. MR 0000390
J. Brillhart, J. Tonascia, and P. Weinberger, On the Fermat quotient, Computers in number theory (Proc. Sci. Res. Council Atlas Sympos. No. 2, Oxford, 1969) Academic Press, London, 1971, pp. 213–222. MR 0314736
Brown, R. and McIntosh, R. (2001). http://www.loria.fr/~zimmerma/records/Wieferich.status.
Richard Crandall, Karl Dilcher, and Carl Pomerance, A search for Wieferich and Wilson primes, Math. Comp. 66 (1997), no. 217, 433–449. MR 1372002, DOI 10.1090/S0025-5718-97-00791-6
Richard Crandall and Carl Pomerance, Prime numbers, Springer-Verlag, New York, 2001. A computational perspective. MR 1821158, DOI 10.1007/978-1-4684-9316-0
Crump, J. (2002). http://www.spacefire.com/NumberTheory/Wieferich.htm.
Cunningham, A. (1910). Proceedings of the London Mathematical Society, 2(8):xiii.
Fröberg, C.-E. (1958). Some computations of Wilson and Fermat remainders. Mathematical Tables and other Aids to Computation, page 281.
Granlund, T. (2000). GNU MP: The GNU Multiple Precision Arithmetic Library, 3.1.1 edition.
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
Grave, D. (1909). An elementary text on the theory of numbers (in Russian). Kiev Izv. Univ., Kiev.
Haußner, M. and Sachs, D. (1963). On the congruence $2^p \equiv 2 \pmod {p^2}$. American Mathematical Monthly, 70:996.
Hertzer, H. (1908). Über die Zahlen der Form $a^{p-1}-1$, wenn $p$ eine Primzahl. Archiv der Mathematik und Physik, (13):107.
Jacobi, C. and Busch (1828). Journal für die reine und angewandte Mathematik, 3:301–302.
Keller, W. and Richstein, J. (2001). Solutions of the congruence $a^{p-1}\equiv 1\pmod {p^r}$. To appear.
K. E. Kloss, Some number-theoretic calculations, J. Res. Nat. Bur. Standards Sect. B 69B (1965), 335–336. MR 190057
Sidney Kravitz, The congruence $2^{p-1}\equiv 1(\textrm {mod}p^{2})$ for $p<100,000$, Math. Comp. 14 (1960), 378. MR 121334, DOI 10.1090/S0025-5718-1960-0121334-7
D. H. Lehmer, On Fermat’s quotient, base two, Math. Comp. 36 (1981), no. 153, 289–290. MR 595064, DOI 10.1090/S0025-5718-1981-0595064-5
Meissner, W. (1913). Über die Teilbarkeit von $2^{p-1}-2$ durch das Quadrat der Primzahl $p=1093$. Sitzungsberichte, pages 663–667.
Peter L. Montgomery, New solutions of $a^{p-1}\equiv 1\pmod {p^2}$, Math. Comp. 61 (1993), no. 203, 361–363. MR 1182246, DOI 10.1090/S0025-5718-1993-1182246-5
Erna H. Pearson, On the congruences $(p-1)!\equiv -1$ and $2^{p-1}\equiv 1\,(\textrm {mod}\,p^{2})$, Math. Comp. 17 (1964), 194–195. MR 159780, DOI 10.1090/S0025-5718-1963-0159780-0
Paulo Ribenboim, The new book of prime number records, Springer-Verlag, New York, 1996. MR 1377060, DOI 10.1007/978-1-4612-0759-7
Jörg Richstein, Verifying the Goldbach conjecture up to $4\cdot 10^{14}$, Math. Comp. 70 (2001), no. 236, 1745–1749. MR 1836932, DOI 10.1090/S0025-5718-00-01290-4
Hans Riesel, Note on the congruence $a^{p-1}=1$ $(\textrm {mod}$ $p^{2})$, Math. Comp. 18 (1964), 149–150. MR 157928, DOI 10.1090/S0025-5718-1964-0157928-6
Jiro Suzuki, On the generalized Wieferich criteria, Proc. Japan Acad. Ser. A Math. Sci. 70 (1994), no. 7, 230–234. MR 1303569
Wieferich, A. (1909). Zum letzten Fermat’schen Theorem. Journal für die reine und angewandte Mathematik, 136:293–302.