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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

Puni tekst: engleski, pdf (204 KB) str. 61-79 preuzimanja: 255* citiraj
APA 6th Edition
Gusic, I. (2012). Some applications of the abc-conjecture to the diophantine equation qym=f(x). Glasnik matematički, 47 (1), 61-79. https://doi.org/10.3336/gm.47.1.05
MLA 8th Edition
Gusic, Ivica. "Some applications of the abc-conjecture to the diophantine equation qym=f(x)." Glasnik matematički, vol. 47, br. 1, 2012, str. 61-79. https://doi.org/10.3336/gm.47.1.05. Citirano 18.10.2021.
Chicago 17th Edition
Gusic, Ivica. "Some applications of the abc-conjecture to the diophantine equation qym=f(x)." Glasnik matematički 47, br. 1 (2012): 61-79. https://doi.org/10.3336/gm.47.1.05
Harvard
Gusic, I. (2012). 'Some applications of the abc-conjecture to the diophantine equation qym=f(x)', Glasnik matematički, 47(1), str. 61-79. https://doi.org/10.3336/gm.47.1.05
Vancouver
Gusic I. Some applications of the abc-conjecture to the diophantine equation qym=f(x). Glasnik matematički [Internet]. 2012 [pristupljeno 18.10.2021.];47(1):61-79. https://doi.org/10.3336/gm.47.1.05
IEEE
I. Gusic, "Some applications of the abc-conjecture to the diophantine equation qym=f(x)", Glasnik matematički, vol.47, br. 1, str. 61-79, 2012. [Online]. https://doi.org/10.3336/gm.47.1.05

Sažetak
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.

Ključne riječi
abc-conjecture; Diophantine equation

Hrčak ID: 82571

URI
https://hrcak.srce.hr/82571

Posjeta: 470 *