Proceedings of the Japan Academy, Series A, Mathematical Sciences

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

Zhenfu Cao, Xiaolei Dong, and Zhong Li

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Abstract

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 $(m + \sqrt{-1} )^r = V_r + U_r \sqrt{-1}$.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 78, Number 10 (2002), 199-202.

Dates
First available in Project Euclid: 23 May 2006

Permanent link to this document
https://projecteuclid.org/euclid.pja/1148392271

Digital Object Identifier
doi:10.3792/pjaa.78.199

Mathematical Reviews number (MathSciNet)
MR1950170

Zentralblatt MATH identifier
1093.11022

Subjects
Primary: 11D61: Exponential equations

Keywords
Exponential Diophantine equation Lucas sequence primitive divisor Gauss integer

Citation

Cao, Zhenfu; Dong, Xiaolei; Li, Zhong. A new conjecture concerning the Diophantine equation $x^2 + b^y = c^z$. Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), no. 10, 199--202. doi:10.3792/pjaa.78.199. https://projecteuclid.org/euclid.pja/1148392271


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References

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