On the generalized Ramanujan--Nagell equation $D_1x^2-D_2=p^n$
Abstract
Let $D$ be a nonsquare positive integer, and let $D_1$, $D_2$ be positive integers such that $D_1D_2=D$ and $\gcd(D_1,D_2)=1$. Further, let $p$ be an odd prime with $p\nmid D$. Denote by $N(D_1,D_2,p)$ the number of positive integer solutions $(x,n)$ to the generalized Ramanujan--Nagell equation $D_1x^2-D_2=p^n$. In this paper, we prove $N(D_1,D_2,p)\le4$. Moreover, we generalize the case where $D_1=1$ to give the {\it exceptional} triple $(D_1,D_2,p)$, which satisfies $$p^{\nu}=D_1a^2+\eta,\quad D_2=\dfrac{1}{D_1}\left(\dfrac{p^m-\eta}{2a}\right)^2-p^m, \qquad a,\nu,m\in\mathbb{N},\quad \nu
Fields of Study & Subjects
Algebraic Geometry and Number Theory
Analytic Number Theory Research
Advanced Mathematical Identities
Publication details
- Journal / source
- Publicationes Mathematicae Debrecen
- Volume
- 108
- Issue
- 1-2
- Pages
- 221–250
- Publication date
- 1 January 2026