which holds for allodd 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." http://elmath.org/.Goldfeld,
D. "Modular Forms, Elliptic Curves and the -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. Canada8, 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. http://www.loria.fr/~zimmerma/records/Wieferich.status.Montgomery,
P. "New Solutions of ." 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.
which is a solution to the congruence
equation
(Lehmer 1981, Crandall 1986, Crandall et al.
1997), a limit since increased to 
(McIntosh 2004) and subsequently to 
by PrimeGrid as of November 2015.
,
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).
-Conjecture." 
)."
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. 
." Math. Comput. 61, 361-363,
1991.PrimeGrid PRPNet. "Wieferich Prime Search."