Wieferich Prime

A Wieferich prime is a Prime $p$ which is a solution to the Congruence equation

$$2^{p-1}\equiv 1\ \left({{\rm mod\ } {p^2}}\right).$$

Note the similarity of this expression to the special case of Fermat's Little Theorem

$$2^{p-1}\equiv 1\ \left({{\rm mod\ } {p}}\right),$$

which holds for all Odd Primes. However, the only Wieferich primes less than $4\times 10^{12}$ are $p=1093$ and 3511 (Lehmer 1981, Crandall 1986, Crandall et al. 1997). Interestingly, one less than these numbers have suggestive periodic Binary representations

$$1092 = 10001000100_2$$

$$3510 = 110110110110_2.$$

A Prime factor $p$ of a Mersenne Number $M_q=2^q-1$ is a Wieferich prime Iff $p^2\vert 2^q-1$. 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 $p\vert 2^n\pm 1$ with $p$ and $n$ Relatively Prime, then $p$ is a Wieferich prime Iff $p^2$ also divides $2^n\pm 1$. The Conjecture that there are no three Powerful Numbers implies that there are infinitely many Wieferich primes (Granville 1986, Vardi 1991). In addition, the abc Conjecture implies that there are at least $C\ln x$ Wieferich primes $\leq x$ for some constant $C$ (Silverman 1988, Vardi 1991).

See also abc Conjecture, 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.

Granville, A. ``Powerful Numbers and Fermat's Last Theorem.'' C. R. Math. Rep. Acad. Sci. Canada 8, 215-218, 1986.

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

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.

Vardi, I. ``Wieferich.'' §5.4 in Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 59-62 and 96-103, 1991.

Wieferich, A. ``Zum letzten Fermat'schen Theorem.'' J. reine angew. Math. 136, 293-302, 1909.