A generalization of the Ramanujan-Nagell equation

TL;DR

This paper extends the Ramanujan-Nagell equation to a broader class involving multiple primes and demonstrates that such equations have a finite, explicitly bounded number of solutions.

Contribution

It provides a generalization of the Ramanujan-Nagell equation for arbitrary positive integers and primes, establishing a uniform upper bound on the number of solutions.

Findings

At most 63 solutions for the generalized equation.

Solutions are finite and explicitly bounded.

Applicable to equations with specified prime conditions.

Abstract

We shall show that, for any positive integer $D>0$ and any primes $p_1, p_2$ not dividing $D$, the diophantine equation $x^2+D=2^s p_1^k p_2^l$ has at most $63$ integer solutions $(x, k, l, s)$ with $x, k, l\geq 0$ and $s\in \{0, 2\}$.