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Wieferich Prime


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


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


which holds for all odd primes. The first few Wieferich primes are 1093, 3511, ... (Sloane's A001220), with none other less than (Lehmer 1981, Crandall 1986, Crandall et al. 1997). Interestingly, one less than these numbers have suggestive periodic binary representations

 
 

A prime factor p of a Mersenne number is a Wieferich prime iff . Therefore, Mersenne primes are not Wieferich primes.

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

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




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.

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.

Lehmer, D. H. "On Fermat's Quotient, Base Two." Math. Comput. 36, 289-290, 1981.

Montgomery, P. "New Solutions of ." Math. Comput. 61, 361-363, 1991.

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. Sequences A001220 in "The On-Line Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/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.