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

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 + . . . + ( 2 x ) 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.