A note on non-integrality of the $(k,l)$-G\"{o}bel sequences

Yuh Kobayashi; Shin-ichiro Seki

A note on non-integrality of the $(k,l)$-G\"{o}bel sequences

Number Theory 2024-12-24 v3

Authors: Yuh Kobayashi , Shin-ichiro Seki

Abstract

The $(k,l)$ -G\"{o}bel sequences defined by Ibstedt remain integers for the first (in some cases, many) terms, but for selected values of $(k,l)$ , computations show that the terms eventually stop being integers. It is still unresolved whether the integrality of these sequences breaks down for all $k, l\geq 2$ . In this article, we prove the non-integrality for a specific class of $(k,l)$ values. Our proof is based on geometric arguments related to the distribution of quadratic residues modulo a prime.

Keywords

G"{o}bel sequences non-integrality quadratic residues modular arithmetic number theory

Cite

@article{arxiv.2410.23240,
  title  = {A note on non-integrality of the $(k,l)$-G\"{o}bel sequences},
  author = {Yuh Kobayashi and Shin-ichiro Seki},
  journal= {arXiv preprint arXiv:2410.23240},
  year   = {2024}
}

Comments

20 pages, 5 figures, Only the introduction to the paper is changed; there is no change to its mathematical content

A note on non-integrality of the $(k,l)$-G\"{o}bel sequences

Abstract

Keywords

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