abc Conjecture

A Conjecture due to J. Oesterlé and D. W. Masser. It states that, for any Infinitesimal $\epsilon>0$, there exists a Constant $C_\epsilon$ such that for any three Relatively Prime Integers $a$, $b$, $c$ satisfying

$$a+b=c,$$

the Inequality

$$\mathop{\rm max} \{\vert a\vert,\vert b\vert,\vert c\vert\} \leq C_\epsilon \prod_{p\vert abc} p^{1+\epsilon}$$

holds, where $p\vert abc$ indicates that the Product is over Primes $p$ which Divide the Product $abc$. If this Conjecture were true, it would imply Fermat's Last Theorem for sufficiently large Powers (Goldfeld 1996). This is related to the fact that the abc conjecture implies that there are at least $C\ln x$ Wieferich Primes $\leq x$ for some constant $C$ (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.