Available in electronic format
Available in print format		 Mathematics of Computation
Journal of the American Mathematical Society
ISSN 1088-6842(e) ISSN 0025-5718(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

The continuing search for Wieferich primes

Author(s): Joshua Knauer; Jörg Richstein.
Journal: Math. Comp. 74 (2005), 1559-1563.
MSC (2000): Primary 11A07; Secondary 11-04
Posted: January 19, 2005
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

Abstract: A prime $p$ satisfying the congruence

\begin{displaymath}2^{p-1} \equiv 1 \pmod{p^2}\end{displaymath}

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:

[1]
Abel, N. H. (1828). Journal für die reine und angewandte Mathematik, 3:212.

[2]
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.

[3]
Beeger, N. (1922). On a new case of the congruence $2^{p-1} \equiv 1 \pmod{p^2}$. Messenger of Mathematics, 51:149-150.

[4]
Beeger, N. (1940). On the congruence $2^{p-1} \equiv 1 \pmod{p^2}$ and Fermat's last theorem. Nieuw Archief Voor Wiskunde, pages 51-54. MR 0000390 (1:65d)

[5]
Brillhart, J., Tonascia, J., and Weinberger, P. (1971). On the Fermat quotient. Computers in Number Theory, pages 213-222. MR 0314736 (47:3288)

[6]
Brown, R. and McIntosh, R. (2001). http://www.loria.fr/ ~zimmerma/records/Wieferich.status.

[7]
Crandall, R., Dilcher, K., and Pomerance, C. (1997). A search for Wieferich and Wilson primes. Mathematics of Computation, 66(217):433-489. MR 1372002 (97c:11004)

[8]
Crandall, R. and Pomerance, C. (2001). Prime Numbers - A Computational Perspective. Springer-Verlag, New York, 2001. MR 1821158 (2002a:11007)

[9]
Crump, J. (2002). http://www.spacefire.com/NumberTheory/Wieferich.htm.

[10]
Cunningham, A. (1910). Proceedings of the London Mathematical Society, 2(8):xiii.

[11]
Fröberg, C.-E. (1958). Some computations of Wilson and Fermat remainders. Mathematical Tables and other Aids to Computation, page 281.

[12]
Granlund, T. (2000). GNU MP: The GNU Multiple Precision Arithmetic Library, 3.1.1 edition.

[13]
Granville, A. and Monagan, M. B. (1988). The first case of Fermat's last theorem is true for all prime exponents up to 714,591,416,091,389. Transactions of the American Mathematical Society, (306):329-359. MR 0927694 (89g:11025)

[14]
Grave, D. (1909). An elementary text on the theory of numbers (in Russian). Kiev Izv. Univ., Kiev.

[15]
Haußner, M. and Sachs, D. (1963). On the congruence $2^p \equiv 2 \pmod{p^2}$. American Mathematical Monthly, 70:996.

[16]
Hertzer, H. (1908). Über die Zahlen der Form $a^{p-1}-1$, wenn $p$ eine Primzahl. Archiv der Mathematik und Physik, (13):107.

[17]
Jacobi, C. and Busch (1828). Journal für die reine und angewandte Mathematik, 3:301-302.

[18]
Keller, W. and Richstein, J. (2001). Solutions of the congruence $a^{p-1}\equiv 1\pmod {p^r}$. To appear.

[19]
Kloss, K. E. (1965). Some number-theoretic calculations. Journal of Research of the National Bureau of Standards-B. Mathematics and Mathematical Physics, 69B(4):335-336. MR 0190057 (32:7473)

[20]
Kravitz, S. (1960). The congruence $2^{p-1} \equiv 1 \pmod{p^2}$ for $p < 100,000$. Mathematics of Computation, page 378. MR 0121334 (22:12073)

[21]
Lehmer, D. (1981). On Fermat's quotient, base two. Mathematics of Computation, 36(153):289-290. MR 0595064 (82e:10004)

[22]
Meissner, W. (1913). Über die Teilbarkeit von $2^{p-1}-2$ durch das Quadrat der Primzahl $p=1093$. Sitzungsberichte, pages 663-667.

[23]
Montgomery, P. L. (1993). New prime solutions of $a^{p-1} \equiv 1 \pmod{p^2}$. Mathematics of Computation, 203(61):361-363. MR 1182246 (94d:11003)

[24]
Pearson, E. H. (1963). On the congruences $(p-1)! \equiv -1$ and $2^{p-1} \equiv 1 \pmod{p^2}$. Mathematics of Computation, pages 194-195. MR 0159780 (28:2996)

[25]
Ribenboim, P. (1996). The New Book of Prime Number Records. Springer-Verlag, New York, 1996. MR 1377060 (96k:11112)

[26]
Richstein, J. (2000). Verifying the Goldbach conjecture up to $4 \cdot 10^{14}$. Mathematics of Computation, 70(236):1745-1749. MR 1836932 (2002c:11131)

[27]
Riesel, H. (1964). Note on the congruence $a^{p-1} \equiv 1 \pmod{p^2}$. Mathematics of Computation, pages 149-150. MR 0157928 (28:1156)

[28]
Suzuki, J. (1994). On the generalized Wieferich criteria. Proc. Japan Acad. Ser. A Math. Sci., (70):230-234. MR 1303569 (95j:11026)

[29]
Wieferich, A. (1909). Zum letzten Fermat'schen Theorem. Journal für die reine und angewandte Mathematik, 136:293-302.

Similar Articles:

Retrieve articles in Mathematics of Computation with MSC (2000): 11A07, 11-04

Retrieve articles in all Journals with MSC (2000): 11A07, 11-04

Additional Information:

Joshua Knauer
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, V5A 1S6 Canada
Email: jknauer@cecm.sfu.ca

Jörg Richstein
Affiliation: Institut für Informatik, Justus-Liebig-Universität, Gieß{}en, Germany
Email: Joerg.Richstein@informatik.uni-giessen.de

DOI: 10.1090/S0025-5718-05-01723-0
PII: S 0025-5718(05)01723-0
Received by editor(s): June 18, 2003
Received by editor(s) in revised form: April 11, 2004
Posted: January 19, 2005
Additional Notes: The second author was supported in part by the Killam Trusts.
Copyright of article: Copyright 2005, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google