On the Diophantine equation 1k+2k +...+ xk = yn

In this paper, we resolve a conjecture of Schaffer on the solvability of Diophantine equations of the shape 1(k) + 2(k) +... + x(k) = y(n), for 1 less than or equal to k less than or equal to 11. Our method, which may, with a modicum of effort, be extended to higher values of k, combines a wide variety of techniques, classical and modern, in Diophantine analysis.