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Schäffer's conjecture for even k ≤ 42
Abstract
In 1956, Schäffer made a conjecture on the whole number solutions to the Diophantine equation Sk(x):=1k+2k+⋯+xk=yn. In this paper we show how for even k, we can descend to a long list of three coupled equations of the form Axp−Byp=1, where A,B∈ℚ. To this system, we apply the modular method, linear forms in logarithms, and local methods. As a result, we prove Schäffer's conjecture for even k ≤ 42.
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BibTeX
@misc{chen2026schäffers,
title = {{Schäffer's conjecture for even $k \le 42$}},
author = {Imin Chen and Stephen Choi and James Houle},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28768},
eprint = {2610.28768},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28768},
}
RIS
TY - UNPB
TI - Schäffer's conjecture for even k ≤ 42
AU - Imin Chen
AU - Stephen Choi
AU - James Houle
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28768
UR - https://arxiv.org/abs/2610.28768
AB - In 1956, Schäffer made a conjecture on the whole number solutions to the Diophantine equation S_k(x):=1^k+2^k+⋯+x^k=y^n. In this paper we show how for even k, we can descend to a long list of three coupled equations of the form Ax^p−By^p=1, where A,B∈ℚ. To this system, we apply the modular method, linear forms in logarithms, and local methods. As a result, we prove Schäffer's conjecture for even k ≤ 42.
ER -