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Complete solutions of a Lebesgue-Ramanujan-Nagell type equation

Priyanka Baruah, Anup Das, Azizul Hoque

Abstract

We consider the Lebesgue-Ramanujan-Nagell type equation $x^2+5^a13^b17^c=2^m y^n$, where $a,b,c, m\geq 0, n \geq 3$ and $x, y\geq 1$ are unknown integers with $\gcd(x,y)=1$. We determine all integer solutions to the above equation. The proof depends on the classical results of Bilu, Hanrot and Voutier on primitive divisors in Lehmer sequences, and finding all $S$-integral points on a class of elliptic curves.

Complete solutions of a Lebesgue-Ramanujan-Nagell type equation

Abstract

We consider the Lebesgue-Ramanujan-Nagell type equation , where and are unknown integers with . We determine all integer solutions to the above equation. The proof depends on the classical results of Bilu, Hanrot and Voutier on primitive divisors in Lehmer sequences, and finding all -integral points on a class of elliptic curves.

Paper Structure

This paper contains 4 sections, 7 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. The case: $4\mid n$
  3. The case: $n\geq 3$ is prime
  4. Proof of Theorem \ref{['thm']}

Key Result

Theorem 1.1

If $n\ne 3,4,6,12$, then eqn1 has no integer solutions. In case of $n=3,4,6,12$, the integer solutions $(x,y,a,b,c,m)$ are given below.

Theorems & Definitions (9)

  • Theorem 1.1
  • Corollary 1.1
  • Corollary 1.2
  • Proposition 2.1
  • proof
  • Proposition 3.1
  • Lemma 3.1
  • Lemma 3.2
  • proof : Proof of Proposition \ref{['propp']}