Some experiments with Ramanujan-Nagell type Diophantine equations

TL;DR

This paper explores the construction of Ramanujan-Nagell type Diophantine equations with multiple solutions, providing new examples and conjectures based on extensive numerical analysis.

Contribution

It demonstrates the existence of infinite families of such equations with at least four solutions and constructs specific examples with up to six solutions, advancing understanding of their solution structures.

Findings

Existence of infinite sets of equations with at least four solutions.

Construction of equations with exactly five solutions for certain k.

New examples of equations with six solutions in positive integers.

Abstract

Stiller proved that the Diophantine equation $x^2+119=15\cdot 2^{n}$ has exactly six solutions in positive integers. Motivated by this result we are interested in constructions of Diophantine equations of Ramanujan-Nagell type $x^2=Ak^{n}+B$ with many solutions. Here, $A,B\in\Z$ (thus $A, B$ are not necessarily positive) and $k\in\Z_{\geq 2}$ are given integers. In particular, we prove that for each $k$ there exists an infinite set $\cal{S}$ containing pairs of integers $(A, B)$ such that for each $(A,B)\in \cal{S}$ we have $\gcd(A,B)$ is square-free and the Diophantine equation $x^2=Ak^n+B$ has at least four solutions in positive integers. Moreover, we construct several Diophantine equations of the form $x^2=Ak^n+B$ with $k>2$, each containing five solutions in non-negative integers. %For example the equation $y^2=130\cdot 3^{n}+5550606$ has exactly five solutions with $n=0, 6, 11, 15, 16$. We also find new examples of equations $x^2=A2^{n}+B$ having six solutions in positive integers, e.g. the following Diophantine equations has exactly six solutions: \begin{equation*} \begin{array}{ll} x^2= 57\cdot 2^{n}+117440512 & n=0, 14, 16, 20, 24, 25, x^2= 165\cdot 2^{n}+26404 & n=0, 5, 7, 8, 10, 12. \end{array} \end{equation*} Moreover, based on an extensive numerical calculations we state several conjectures on the number of solutions of certain parametric families of the Diophantine equations of Ramanujan-Nagell type.