On the Diophantine equation (x + 1)k + (x + 2)k + ... + (2x)k = yn

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 Tk(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. [6]), we show that for 2≤x≤13,k≥1,y≥2 and n≥3 the title equation has no solutions.