abc Conjecture - Some Consequences

Some Consequences

The abc conjecture has a large number of consequences. These include both known results (some of which have been proven separately since the conjecture has been stated), and conjectures for which it gives a conditional proof. While an earlier proof of the conjecture would have been more significant in terms of consequences, the abc conjecture itself remains of interest for the other conjectures it would prove, together with its numerous links with deep questions in number theory.

• Thue–Siegel–Roth theorem on diophantine approximation of algebraic numbers
• Fermat's Last Theorem for all sufficiently large exponents (already proven in general by Andrew Wiles) (Granville 2002)
• The Mordell conjecture (already proven in general by Gerd Faltings) (Elkies 1991)
• The Erdős–Woods conjecture except for a finite number of counterexamples (Langevin 1993)
• The existence of infinitely many non-Wieferich primes (Silverman 1988)
• The weak form of Marshall Hall's conjecture on the separation between squares and cubes of integers (Nitaj 1996)
• The Fermat–Catalan conjecture, a generalization of Fermat's last theorem concerning powers that are sums of powers (Pomerance 2008)
• The L function L(s,(−d/.)) formed with the Legendre symbol, has no Siegel zero (this consequence actually requires a uniform version of the abc conjecture in number fields, not only the abc conjecture as formulated above for rational integers) (Granville 2000)
• P(x) has only finitely many perfect powers for integral x for P a polynomial with at least three simple zeros.
• A generalization of Tijdeman's theorem concerning the number of solutions of (Tijdeman's theorem answers the case ), and Pillai's conjecture (1931) concerning the number of solutions of
• It is equivalent to the Granville–Langevin conjecture.
• It is equivalent to the modified Szpiro conjecture, which would yield a bound of (Oesterlé 1988).
• Dąbrowski (1996) has shown that the abc conjecture implies that n! + A= k2 has only finitely many solutions for any given integer A.