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Contributions to some conjectures on a ternary exponential Diophantine equation

Volume 186 / 2018

Takafumi Miyazaki Acta Arithmetica 186 (2018), 1-36 MSC: Primary 11D61; Secondary 11J86, 11D41. DOI: 10.4064/aa8656-2-2018 Published online: 12 October 2018

Abstract

We consider the exponential Diophantine equation $a^x+b^y=c^z$ for given pairwise coprime positive integers $a$, $b$ and $c$. The case where both $x$ and $y$ are even is thoroughly studied in order to solve the equation for some infinite classes of triples $(a,b,c)$ which have been unassailable via the previously existing methods. It may be recognized from one of our results that a “quarter” of a major unsolved problem of Jeśmanowicz concerning primitive Pythagorean triples is almost solved.

Authors

  • Takafumi MiyazakiDivision of Pure and Applied Science
    Faculty of Science and Technology
    Gunma University
    1-5-1 Tenjin-cho
    Kiryu, Gunma, Japan
    e-mail

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