Multiple Solutions to Exponential Diophantine Equations of Ramanujan-Nagell type: $Cz^2 = D + A.B^n$

TL;DR

This paper investigates exponential Diophantine equations of Ramanujan-Nagell type, identifying cases with multiple solutions and presenting new findings, including a specific equation with four solutions and a conjecture about its uniqueness.

Contribution

The paper provides a comprehensive analysis of Ramanujan-Nagell type equations, discovering new solutions and proposing conjectures about their uniqueness.

Findings

Equation $z^2=277665.17^6+34^n$ has four solutions.

Summaries of multiple solutions for various cases are provided.

Conjecture that the specific equation with four solutions is unique under certain conditions.

Abstract

Ramanujan found five solutions to the exponential Diophantine equation $z^2+7= 2^n$ where $z$ and $n$ are positive integers, and posed the problem of determining whether there are any more. Nagell was the first to prove that there were not. It is natural to then ask if there are other similar Diophantine equations with multiple solutions. In particular, equations of the form $Cz^2=D+A.B^n$ are known as equations of Ramanujan-Nagell type. Three examples are known with six solutions. Summaries of multiple solutions for different cases are presented. In particular the equation $z^2=277665.17^6+34^n$ is found to have four solutions and is conjectured to be the only such non-trivial equation of Ramanujan-Nagell type with four solutions when $A=C=1$ and $B$ is not a power of 2.