abc Conjecture
A Conjecture due to J. Oesterlé and D. W. Masser. It states that, for any Infinitesimal 
,there exists a Constant 
such that for any three Relatively PrimeIntegers 
, 
, 
satisfying
the Inequality
holds, where
indicates that the Product is over Primes 
which Divide the Product 
. If this Conjecture were true, it would imply Fermat's Last Theorem for sufficiently largePowers (Goldfeld 1996). This is related to the fact that the abc conjecture implies that there are atleast 
Wieferich Primes 
for some constant 
(Silverman 1988, Vardi 1991).See also Fermat's Last Theorem, Mason's Theorem, Wieferich Prime
References
Cox, D. A. ``Introduction to Fermat's Last Theorem.'' Amer. Math. Monthly 101, 3-14, 1994. Goldfeld, D. ``Beyond the Last Theorem.'' The Sciences, 34-40, March/April 1996. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 75-76, 1994. Silverman, J. ``Wieferich's Criterion and the abc Conjecture.'' J. Number Th. 30, 226-237, 1988. Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, p. 66, 1991.
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