Jiayuan Hu, Xiaoxue Li : On the generalized Ramanujan-Nagell equation $x^2+q^m=c^n$ with $q^r+1=2c^2$, 257-265

Abstract:

Let $q$ be an odd prime, and let $c$, $r$ be positive integers with $q^r+1=2c^2$. For any nonnegative integer $s$, let $U_{2s+1}=\left(\alpha^{2s+1}+\beta^{2s+1}\right)/2$ and $V_{2s+1}=\left(\alpha^{2s+1}-\beta^{2s+1}\right)/2\sqrt{2}$, where $\alpha=1+\sqrt{2}$ and $\beta=1-\sqrt{2}$. In this paper we prove the following results: (i) If $r>2$, then $(q,r,c)=(23,3,78)$ and the equation $x^2+23^m=78^n$ has only the positive integer solution $(x,m,n)=(6083,3,4)$. (ii) If $r=2$ and $(q,c)=(U_{2s+1},V_{2s+1})$ with $s\not\equiv0(\bmod 4)$, then the equation $x^2+q^m=c^n$ has only the positive integer solution $(x,m,n)=(c^2-1,2,4)$.

Key Words: exponential diophantine equation; generalized Ramanujan-Nagell equation; Pell number.

2010 Mathematics Subject Classification: 11D61.

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