Some applications of the abc-conjecture to the diophantine equation qy^m = f(x) (CROSBI ID 179329)
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Gusić, Ivica
engleski
Some applications of the abc-conjecture to the diophantine equation qy^m = f(x)
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.
abc-conjecture; diophantine equation
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