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Le, Maohua

A note on the diophantine equation \(x^{2}+b^{y}=c^{z}\). (English) Zbl 1164.11319

Czech. Math. J. 56, No. 4, 1109-1116 (2006).
Summary: Let \(a\), \(b\), \(c\), \(r\) be positive integers such that \(a^{2}+b^{2}=c^{r}\), \(\min (a,b,c,r)>1\), \(\text{gcd}(a,b)=1\), \(a\) is even and \(r\) is odd. In this paper we prove that if \(b\equiv 3\pmod 4\) and either \(b\) or \(c\) is an odd prime power, then the equation \(x^{2}+b^{y}=c^{z}\) has only the positive integer solution \((x,y,z)=(a,2,r)\) with \(\min (y,z)>1\).

Cited in 2 Documents

MSC:

11D61 Exponential Diophantine equations

Keywords:

exponential diophantine equation; Lucas number; positive divisor
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\textit{M. Le}, Czech. Math. J. 56, No. 4, 1109--1116 (2006; Zbl 1164.11319)
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References:

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