Erdős-Moser Equation in Arithmetic Progressions

Anji Dong; Vi Anh Nguyen; Alexandru Zaharescu

Erdős-Moser Equation in Arithmetic Progressions

Number Theory 2026-08-04 v1

Authors: Anji Dong , Vi Anh Nguyen , Alexandru Zaharescu

Abstract

We consider the Erd\H{o}s-Moser equation $1^k+2^k+\cdots+(m-1)^k=m^k$ in arithmetic progressions. We prove among other things that when $k=2$ , for any solution to exist, the above sum in arithmetic progression must consist of two or four terms. In either case, there are infinitely many solutions that can be completely characterized.

Cite

@article{arxiv.2608.03375,
  title  = {Erdős-Moser Equation in Arithmetic Progressions},
  author = {Anji Dong and Vi Anh Nguyen and Alexandru Zaharescu},
  journal= {arXiv preprint arXiv:2608.03375},
  year   = {2026}
}

Comments

14 pages, 3 figures