Glasnik Matematicki, Vol. 47, No. 1 (2012), 61-79.

SOME APPLICATIONS OF THE ABC-CONJECTURE TO THE DIOPHANTINE EQUATION QY^M=F(X)

Ivica Gusić

Abstract.   Assume that the abc-conjecture is true. Let f be a polynomial over Q of degree n≥ 2 and let m≥ 2 be an integer such that the curve y^m=f(x) has genus ≥ 2. A. Granville in [3] proved that there is a set of exceptional pairs (m,n) such that if (m,n) is not exceptional, then the equation dy^m=f(x) has only trivial rational solutions, for almost all m-free integers d. We prove that the result can be partially extended on the set of exceptional pairs. For example, we prove that if f is completely reducible over Q and n ≠ 2, then the equation qy^m=f(x) has only trivial rational solutions, for all but finitely many prime numbers q.

2010 Mathematics Subject Classification.   11D45, 11D41.

Key words and phrases.   abc-conjecture, Diophantine equation.

DOI: 10.3336/gm.47.1.05

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