On the equation $1^k + 2^k + ... + x^k = y^z$

Polynomial values of (alternating) power sums

“…Then, for k > 3 equation (13) has only finitely many integer solutions x, y, unless one of the following holds:…”

“…x > 2) solution of equation (4) is (k, n, x, y) = (2, 2, 24, 70). Győry, Tijdeman and Voorhoeve [13] proved effective finiteness for the solutions of (4) in the general case when, in (4), n is also unknown. Several generalizations of (4) have been considered, e.g.…”

“…So, if, in our case, equation (13) has infinitely many solutions, then g(x) is of the form as in Theorem 1(I).…”

Lucas' square pyramid problem revisited

“…Attempts to repair this perceived defect have, in recent years, resulted in a number of elementary proofs, by Ma [26] and [27], Cao and Yu [6], Cucurezeanu [10] and Anglin [2]. Various generalizations, distinct from that considered here, have been addressed in [12] and [33].…”

On multiple zeros of Bernoulli polynomials

“…One question that has been asked is that of the possibility of multiple zeros. While this is interesting in its own right, zero multiplicity properties of Bernoulli and related polynomials have been used in the study of certain diophantine equations involving sums of powers; see, e.g., [10], [15], or [4].…”