Wieferich Prime
A Wieferich prime is a Prime 
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. However, the only Wieferich primes less than
are 
and3511 (Lehmer 1981, Crandall 1986, Crandall et al. 1997). Interestingly, one less than these numbers have suggestive periodicBinary representations |  |  | |
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A Prime factor
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 , then must be a Wieferich prime (Wieferich1909). If 
with and 
Relatively Prime, then is a Wieferich prime Iff 
alsodivides 
. The Conjecture that there are no three Powerful Numbers implies thatthere are infinitely many Wieferich primes (Granville 1986, Vardi 1991). In addition, the abc Conjecture impliesthat there are at least 
Wieferich primes 
for some constant 
(Silverman 1988, Vardi 1991).
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.
- Pseudovector
- Psi Function
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- Ptolemy's Theorem
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