On the equation \(1^k+2^k+\ldots+x^k=y^n\). (English) Zbl 1026.11042
Summary: In our paper a survey is given on the title equation. General finiteness theorems, bounds on \(n\) and the number of solutions, complete solution for small values of \(k\), and some generalizations and analogues are presented. The basic ideas of the most important proofs are also outlined. We note that in the proofs of the recent results virtually every technique of modern Diophantine analysis has been employed.
MSC:
11D61 | Exponential Diophantine equations |
11B68 | Bernoulli and Euler numbers and polynomials |
11D45 | Counting solutions of Diophantine equations |
11-02 | Research exposition (monographs, survey articles) pertaining to number theory |
11D41 | Higher degree equations; Fermat’s equation |