A Wieferich prime is a prime 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 no other example reported below et al.
1997). McIntosh (2004) subsequently reported a search limit of
The numbers one less than these primes have the binary
representations
(Johnson 1977).
If the first case of Fermat's last theorem is false for exponent positive
integer odd prime relatively prime to iff conjecture
that there are no three consecutive powerful numbers
implies that there are infinitely many non-Wieferich primes (Granville 1986, Vardi
1991; Ribenboim 1996, p. 341). In addition, the abc
conjecture implies that there are at least
See also abc Conjecture ,
Double Wieferich Prime Pair ,
Fermat Quotient ,
Fermat's Last Theorem ,
Integer
Sequence Primes ,
Mersenne Number ,
Mirimanoff's
Congruence ,
Powerful Number
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References Brillhart, J.; Tonascia, J.; and Winberger, P. "On the Fermat Quotient." In Computers
and Number Theory Crandall, R. E.
Projects
in Scientific Computation. Crandall,
R.; Dilcher, K.; and Pomerance, C. "A Search for Wieferich and Wilson Primes."
Math. Comput. 66 , 433-449, 1997. https://doi.org/10.1090/S0025-5718-97-00791-6 . Dobeš,
J. "elMath.org: Project Wieferich@Home." https://elmath.org/ . Goldfeld,
D. "Modular Forms, Elliptic Curves and the 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. Hardy,
G. H. and Wright, E. M. Th. 91 in An
Introduction to the Theory of Numbers, 5th ed. 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://members.loria.fr/PZimmermann/records/Wieferich.status . Montgomery,
P. L. "New Solutions of Math. Comput. 61 , 361-363,
1993. https://doi.org/10.1090/S0025-5718-1993-1182246-5 . Numberphile.
"Six Sequences." Jul. 22, 2013. https://www.youtube.com/watch?v=VDD6FDhKCYA . PrimeGrid
PRPNet. "Wieferich Prime Search." https://web.archive.org/web/20151121045353/http://prpnet.primegrid.com:13000/ . Ribenboim,
P. "Wieferich Primes." §5.3 in The
New Book of Prime Number Records. Shanks, D. Solved
and Unsolved Problems in Number Theory, 4th ed. Silverman, J. H. "Wieferich's Criterion and
the abc-Conjecture." J. Number Th. 30 , 226-237, 1988. https://doi.org/10.1016/0022-314X(88)90019-4 . Sloane,
N. J. A. Sequence A001220 in "The
On-Line Encyclopedia of Integer Sequences." Vardi, I. "Wieferich."
§5.4 in Computational
Recreations in Mathematica. Wells, D. The
Penguin Dictionary of Curious and Interesting Numbers. Wieferich, A. "Zum letzten Fermat'schen
Theorem." J. reine angew. Math. 136 , 293-302, 1909. Referenced
on Wolfram|Alpha Wieferich Prime
Cite this as:
Weisstein, Eric W. "Wieferich Prime."
From MathWorld https://mathworld.wolfram.com/WieferichPrime.html
Subject classifications More... Less...
which is a solution to the congruence
equation
(Lehmer 1981, Crandall 1994, Crandall et al.
1997). McIntosh (2004) subsequently reported a search limit of 
. PrimeGrid reported a limit of 
as of November 2015.
,
then 
must be a Wieferich prime (Wieferich 1909). For a positive
integer 
and an odd prime 
relatively prime to 
, if 
, then 
is a Wieferich prime iff 
also divides 
, with the same choice of sign. The conjecture
that there are no three consecutive powerful numbers
implies that there are infinitely many non-Wieferich primes (Granville 1986, Vardi
1991; Ribenboim 1996, p. 341). In addition, the abc
conjecture implies that there are at least 
non-Wieferich primes 
for some constant 
and all sufficiently large 
(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,
1993.