A computational approach for solving y2 =1k + 2k + ... + xk

We present a computational approach for finding all integral solutions of the equation y2 = 1k + 2k + · · ·+xk for even values of k. By reducing this problem to that of finding integral solutions of a certain class of quartic equations closely related to the Pell equations, we are able to apply the powerful computational machinery related to quadratic number fields. Using our approach, we determine all integral solutions for 2 ≤ k ≤ 70 assuming the Generalized Riemann Hypothesis, and for 2 ≤ k ≤ 58 unconditionally.