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Original scientific paper

https://doi.org/10.3336/gm.47.1.05

Some applications of the abc-conjecture to the diophantine equation qym=f(x)

Ivica Gusic ; Faculty of Chemical Engin. and Techn., University of Zagreb, Marulićev trg 19, 10000 Zagreb, Croatia


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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 ym=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 dym=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 qym=f(x) has only trivial rational solutions, for all but finitely many prime numbers q.

Keywords

abc-conjecture, Diophantine equation

Hrčak ID:

82571

URI

https://hrcak.srce.hr/82571

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