Geodesic
A note on the diophantine equation $x^2+b^Y=c^z$
Czechoslovak Mathematical Journal, Tome 56 (2006) no. 4, pp. 1109-1116 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $a$, $b$, $c$, $r$ be positive integers such that $a^{2}+b^{2}=c^{r}$, $\min (a,b,c,r)>1$, $\gcd (a,b)=1, a$ is even and $r$ is odd. In this paper we prove that if $b\equiv 3\hspace{4.44443pt}(\@mod \; 4)$ 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 $\min (y,z)>1$.
Let $a$, $b$, $c$, $r$ be positive integers such that $a^{2}+b^{2}=c^{r}$, $\min (a,b,c,r)>1$, $\gcd (a,b)=1, a$ is even and $r$ is odd. In this paper we prove that if $b\equiv 3\hspace{4.44443pt}(\@mod \; 4)$ 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 $\min (y,z)>1$.
Classification : 11D61
Keywords: exponential diophantine equation; Lucas number; positive divisor 
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Le, Maohua. A note on the diophantine equation $x^2+b^Y=c^z$. Czechoslovak Mathematical Journal, Tome 56 (2006) no. 4, pp. 1109-1116. http://geodesic.mathdoc.fr/item/CMJ_2006_56_4_a2/

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