TOPICS
Search

Wieferich Prime


A Wieferich prime is a prime p which is a solution to the congruence equation

 2^(p-1)=1 (mod p^2).
(1)

Note the similarity of this expression to the special case of Fermat's little theorem

 2^(p-1)=1 (mod p),
(2)

which holds for all odd primes. The first few Wieferich primes are 1093, 3511, ... (OEIS A001220), with none other less than 4×10^(12) (Lehmer 1981, Crandall 1986, Crandall et al. 1997), a limit since increased to 1.25×10^(15) (McIntosh 2004) and subsequently to 4.968543×10^(17) by PrimeGrid as of November 2015.

Interestingly, one less than these numbers have suggestive periodic binary representations

1092=10001000100_2
(3)
3510=110110110110_2
(4)

(Johnson 1977).

If the first case of Fermat's last theorem is false for exponent p, then p must be a Wieferich prime (Wieferich 1909). If p|2^n+/-1 with p and n relatively prime, then p is a Wieferich prime iff p^2 also divides 2^n+/-1. The conjecture that there are no three consecutive powerful numbers implies that there are infinitely many non-Wieferich primes (Granville 1986; Ribenboim 1996, p. 341; Vardi 1991). In addition, the abc conjecture implies that there are at least Clnx non-Wieferich primes <=x for some constant C (Silverman 1988, Vardi 1991).


See also

abc Conjecture, Double Wieferich Prime Pair, Fermat's Last Theorem, Fermat Quotient, Integer Sequence Primes, Mersenne Number, Mirimanoff's Congruence, Powerful Number

Explore with Wolfram|Alpha

References

Brillhart, J.; Tonascia, J.; and Winberger, P. "On the Fermat Quotient." In Computers and Number Theory (Ed. A. O. L. Atkin and B. J. Birch). New York: Academic Press, pp. 213-222, 1971.Crandall, R. Projects in Scientific Computation. New York: Springer-Verlag, 1986.Crandall, R.; Dilcher, K; and Pomerance, C. "A Search for Wieferich and Wilson Primes." Math. Comput. 66, 433-449, 1997.Dobeš, J. "elMath.org: Project Wieferich@Home." http://elmath.org/.Goldfeld, D. "Modular Forms, Elliptic Curves and the ABC-Conjecture." http://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf.Granville, A. "Powerful Numbers and Fermat's Last Theorem." C. R. Math. Rep. Acad. Sci. Canada 8, 215-218, 1986.Guy, R. K. §A3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994.Hardy, G. H. and Wright, E. M. Th. 91 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.Johnson, W. "On the Nonvanishing of Fermat Quotients (mod p)." J. reine angew. Math. 292, 196-200, 1977.Lehmer, D. H. "On Fermat's Quotient, Base Two." Math. Comput. 36, 289-290, 1981.McIntosh, R. email to Paul Zimmermann. 9 Mar 2004. http://www.loria.fr/~zimmerma/records/Wieferich.status.Montgomery, P. "New Solutions of a^(p-1)=1 (mod p^2)." Math. Comput. 61, 361-363, 1991.PrimeGrid PRPNet. "Wieferich Prime Search." http://prpnet.primegrid.com:13000.Ribenboim, P. "Wieferich Primes." §5.3 in The New Book of Prime Number Records. New York: Springer-Verlag, pp. 333-346, 1996.Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 116 and 157, 1993.Silverman, J. "Wieferich's Criterion and the abc Conjecture." J. Number Th. 30, 226-237, 1988.Sloane, N. J. A. Sequence A001220 in "The On-Line Encyclopedia of Integer Sequences."Vardi, I. "Wieferich." §5.4 in Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 59-62 and 96-103, 1991.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 163, 1986.Wieferich, A. "Zum letzten Fermat'schen Theorem." J. reine angew. Math. 136, 293-302, 1909.

Referenced on Wolfram|Alpha

Wieferich Prime

Cite this as:

Weisstein, Eric W. "Wieferich Prime." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WieferichPrime.html

Subject classifications