On a conjecture of Schaffer concerning the equation 1k + ... + xk = yn

We prove Schaffer's conjecture concerning the solutions of the equation in the title under certain assumptions on x, letting the other variables k, 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 S-k(x) and on the explicit solution of polynomial-exponential congruences. (C) 2015 Elsevier Inc. All rights reserved.