Proceedings of the Japan Academy, Series A, Mathematical Sciences

The Diophantine equation $x^2 + b^y = c^z$

Zhenfu Cao and Xiaolei Dong

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Abstract

Let $b$ be an odd prime, $m, r \in \mathbf{N}$ with $2 \mid m$ and $2 \nmid r$, $r > 1$, and define the integers $U_r$, $V_r$ by $(m + \sqrt{-1} )^r = V_r + U_r\sqrt{-1}$. In this paper, we prove that if $a = |V_r|$, $b = |U_r|$, $c = m^2 + 1$, and $b > 8 \cdot 10^6$, $b \equiv 3 \pmod{4}$, then the Diophantine equation $x^2 + b^y = c^z$ has only the positive integer solution $(x, y, z) = (a, 2, r)$.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 77, Number 1 (2001), 1-4.

Dates
First available in Project Euclid: 23 May 2006

Permanent link to this document
https://projecteuclid.org/euclid.pja/1148393138

Digital Object Identifier
doi:10.3792/pjaa.77.1

Mathematical Reviews number (MathSciNet)
MR1812738

Zentralblatt MATH identifier
0987.11020

Subjects
Primary: 11D61: Exponential equations

Keywords
Exponential Diophantine equation

Citation

Cao, Zhenfu; Dong, Xiaolei. The Diophantine equation $x^2 + b^y = c^z$. Proc. Japan Acad. Ser. A Math. Sci. 77 (2001), no. 1, 1--4. doi:10.3792/pjaa.77.1. https://projecteuclid.org/euclid.pja/1148393138


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