On the equation 1k + 2k + ⋯ + xk = yn for fixed x

We provide all solutions of the title equation in positive integers x,k,y,nx,k,y,n with 1≤x<251≤x<25 and n≥3n≥3. For these values of the parameters, our result gives an affirmative answer to a related, classical conjecture of Schäffer. In our proofs we combine several tools: Baker's method (in particular, sharp bounds for the linear combinations of logarithms of two algebraic numbers), polynomial-exponential congruences and computational methods.