On the complete solutions of a generalized Lebesgue-Ramanujan-Nagell equation

TL;DR

This work determines all integer solutions to the generalized Lebesgue-Ramanujan-Nagell equation $x^2+17^k41^\ell59^m=2^\delta y^n$ with $x\ge1$, $y>1$, $\gcd(x,y)=1$, and $n\ge3$, by combining classical primitive-divisor theory for Lehmer sequences with reductions to elliptic and quartic curves and extensive computer search. The authors prove $\delta\ge3$ yields no solutions; for $\delta\in\{0,1,2\}$ they completely enumerate solutions: no $n$ beyond $5$, with $n=3$ and $n=4$ solutions given in Tables Tp3 and Tp4, and a unique $n=5$ solution $(x,y,\lambda,k,\ell,m)=(38,5,1,0,2,0)$. The approach integrates $S$-integer techniques, Lehmer-sequence arguments, and systematic Magma computations to obtain a full classification, along with corollaries for related equations. These results advance understanding of generalized Lebesgue-Ramanujan-Nagell equations and provide explicit, verifiable solution sets for several Liouville-type Diophantine problems.

Abstract

We consider the generalized Lebesgue-Ramanujan-Nagell equation $x^2+17^k41^\ell 59^m=2^δy^n$ in the unknown integers $x\geq 1, y>1,n\geq 3$ and $k, \ell, m\geq 0$ satisfying $\gcd(x,y)=1$. We first find all the integer solutions of the above equation, and then use this result to determine all the integer solutions of some other Lebesgue-Ramanujan-Nagell type equations. Our method uses the classical results of Bilu, Hanrot and Voutier on existence of primitive divisors of Lehmer sequences in combination with number theoretic arguments and computer search.

Theorems & Definitions (14)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof