On the Diophantine equation $(x+1)^{k}+(x+2)^{k}+...+(2x)^{k}=y^n$

TL;DR

This paper establishes upper bounds for the exponent n in a specific Diophantine equation involving sums of powers, using prime factorization analysis and Baker's method, and proves non-existence of solutions for small x and certain parameters.

Contribution

It introduces bounds for n based on prime factorization of the sum and combines Baker's method with polynomial exponential congruences to solve the equation.

Findings

No solutions for 2 ≤ x ≤ 13, k ≥ 1, y ≥ 2, n ≥ 3.

Upper bounds for n depending on prime exponents in the sum.

Method combines prime factorization analysis with Baker's technique.

Abstract

In this work, we give upper bounds for $n$ on the title equation. Our results depend on assertions describing the precise exponents of $2$ and $3$ appearing in the prime factorization of $T_{k}(x)=(x+1)^{k}+(x+2)^{k}+...+(2x)^{k}$. Further, on combining Baker's method with the explicit solution of polynomial exponential congruences (see e.g. BHMP), we show that for $2 \leq x \leq 13, k \geq 1, y \geq 2$ and $n \geq 3$ the title equation has no solutions.