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

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 $2\le x\le 13,k\ge 1,y\ge 2$ and $n\ge 3$ the title equation has no solutions.

26 pages, accepted for publication in Journal of Number Theory (2017)

References in corpus (1)

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

Cited by in corpus (2)

• On perfect powers that are sums of cubes of a three term arithmetic progression
• On perfect powers that are sums of cubes of a five term arithmetic progression