Sum of consecutive powers as a perfect power

TL;DR

This paper investigates the solutions to the equation involving sums of consecutive powers equaling perfect powers, focusing on specific congruence classes of the exponent, and employs advanced number theory techniques.

Contribution

It characterizes all solutions for certain ranges of k and specific prime factor conditions using linear forms in logarithms, modular methods, and Thue equations.

Findings

Only solutions at x=0, -1 for 6 ≤ k ≤ 100 when k ≡ 2 mod 4

Solutions for k with odd prime factors ≡ 3 mod 4 are fully characterized

Advanced number theory methods are used to prove these results

Abstract

In this paper we study the equation $$ x^k + (x+1)^k = y^n,\quad n\geq 3, $$ when $k\equiv 2\pmod{4}$. We prove that the only solutions are for $x=0, -1$ when $6\leq k\leq 100$ or for a $k$ with odd prime factors congruent to $3\pmod{4}$. We use linear forms in logarithms, the modular method and the resolution of Thue equations.