Open AccessSome applications of the abc-conjecture to the diophantine equation $qy^m=f(x)$
Author(s) -
Ivica Gusić
Publication year - 2012
Publication title -
glasnik matematički
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.332
H-Index - 17
eISSN - 1846-7989
pISSN - 0017-095X
DOI - 10.3336/gm.47.1.05
Subject(s) - diophantine equation , mathematics , conjecture , legendre's equation , abc conjecture , diophantine set , thue equation , pure mathematics , beal's conjecture
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