On a conjecture of Schäffer concerning the equation 1k+⋯+xk=yn1k+⋯+xk=yn

We prove Schäffer's conjecture concerning the solutions of the equation in the title under certain assumptions on x  , letting the other variables k,n,yk,n,y be completely free. We also provide upper bounds for n under more moderate conditions. Finally, we give all solutions of the equation in the title for some concrete values of x  . Our results rely on assertions describing the precise exponents of 2 and 3 appearing in the prime factorization of Sk(x)Sk(x) and on the explicit solution of polynomial–exponential congruences.