Wieferich Prime
A Wieferich prime is a prime 
which is a solution
to the congruence equation
 |
(1)
|
Note the similarity of this expression to the special case of Fermat's little theorem
 |
(2)
|
which holds for all odd primes. The first few Wieferich primes are 1093, 3511, ... (OEIS A001220),
with none other less than 
(Lehmer
1981, Crandall 1986, Crandall et al. 1997), a limit since increased to 
(McIntosh 2004) and subsequently
to 
by PrimeGrid as
of November 2015.
Interestingly, one less than these numbers have suggestive periodic binary representations
 |  |  |
(3)
|
 |  |  |
(4)
|
(Johnson 1977).
If the first case of Fermat's last theorem is false for exponent 
, then 
must be a Wieferich
prime (Wieferich 1909). If 
with 
and 
relatively
prime, then 
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; Ribenboim 1996, p. 341; Vardi 1991). In addition, the abc
conjecture implies that there are at least 
non-Wieferich
primes 
for some constant 
(Silverman 1988,
Vardi 1991).
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." https://elmath.org/.
Goldfeld, D. "Modular Forms, Elliptic Curves and the 
-Conjecture."
https://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 
)." 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. https://www.loria.fr/~zimmerma/records/Wieferich.status.
Montgomery, P. "New Solutions of 
."
Math. Comput. 61, 361-363, 1991.
PrimeGrid PRPNet. "Wieferich Prime Search." https://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.
Weisstein, Eric W. "Wieferich Prime." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WieferichPrime.html
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