| Summary: | Abstract Let a be a positive integer with a > 1 $a>1$ , and let ( x , y , n ) $(x, y, n)$ be a positive integer solution of the equation x 2 + a 2 = y n $x^{2}+a^{2}=y^{n}$ , gcd ( x , y ) = 1 $\gcd(x, y)=1$ , n > 2 $n>2$ . Using Baker’s method, we prove that, for any positive number ϵ, if n is an odd integer with n > C ( ϵ ) $n>C(\epsilon)$ , where C ( ϵ ) $C(\epsilon)$ is an effectively computable constant depending only on ϵ, then n < ( 2 + ϵ ) ( log a ) / log y $n<(2+\epsilon)(\log a)/\log y$ . Owing to the obvious fact that every solution ( x , y , n ) $(x, y, n)$ of the equation satisfies n > 2 ( log a ) / log y $n>2(\log a)/\log y$ , the above upper bound is optimal.
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