Yongzhong Hu and Maohua Le: On the number of solutions of the generalized Ramanujan-Nagell equation $D_1x^2+D_2^m=p^n$, p.279-293

Abstract:

Let $D_1$ and $D_2$ be coprime positive integers with $\min(D_1,D_2)>1$, and let $p$ be an odd prime with $p\not\vert D_1D_2$. Further, let $N(D_1,D_2,p)$ denote the number of positive integer solutions $(x,m,n)$ of the equation $D_1x^2+D_2^m=p^n$. In this paper, we prove that $N(D_1,D_2,p)\leq 2$ except for $N(2,7,3)=N(10,3,13)=N(10,3,37)=N((3^{2l-1}-1)/a^2,3,4\cdot3^{2l-1}-1)=3$, where $a,l$ are positive integers.

Key Words: Exponential diophantine equation; generalized Ramanujan-Nagell equation; number of solutions; upper bound.

2000 Mathematics Subject Classification: Primary: 11D61;
Secondary: 11D25.

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