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arXiv · 1709.00400

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

Attila Bérczes·István Pink·Gamze SavaŞ·Gökhan Soydan
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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.

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Attila Bérczes, István Pink, Gamze SavaŞ, Gökhan Soydan. 2017-09-01. On the Diophantine equation $(x+1)^{k}+(x+2)^{k}+...+(2x)^{k}=y^n$. https://doi.org/10.1016/j.jnt.2017.07.020

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