A new conjecture concerning the Diophantine equation $x^2 + b^y = c^z$

Autor:	Zhenfu Cao, Zhong Li, Xiaolei Dong
Rok vydání:	2002
Předmět:	Discrete mathematics 11D61 Conjecture Lehmer number primitive divisor Lucas sequence General Mathematics Diophantine equation Gauss integer Number theory Integer Exponential Diophantine equation Prime power Mathematics
Zdroj:	Proc. Japan Acad. Ser. A Math. Sci. 78, no. 10 (2002), 199-202
ISSN:	0386-2194
DOI:	10.3792/pjaa.78.199
Popis:	In this paper, using a recent result of Bilu, Hanrot and Voutier on primitive divisors, we prove that if $a = |V_r|$, $b = |U_r|$, $c = m^2 + 1$, and $b \equiv 3 \pmod{4}$ is a prime power, then the Diophantine equation $x^2 + b^y = c^z$ has only the positive integer solution $(x,y,z) = (a,2,r)$, where $r > 1$ is an odd integer, $m \in \mathbf{N}$ with $2 \mid m$ and the integers $U_r$, $V_r$ satisfy $(m + \sqrt{-1} )^r = V_r + U_r \sqrt{-1}$.
Databáze:	OpenAIRE
Externí odkaz:	https://explore.openaire.eu/search/publication?articleId=doi_dedup___::7ae2b7a2c44515501f073ecd5d8b93d2 https://doi.org/10.3792/pjaa.78.199 Zobrazit plný text záznamu