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

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 $T_k(x)=(x+1{)}^k+(x+2{)}^k+\mathrm{.}\mathrm{.}\mathrm{.}+(2x{)}^k$. 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.