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

TL;DR

This paper investigates the conditions under which the $(k,l)$-G"obel sequences cease to be integers, providing a proof of non-integrality for certain parameter values using geometric and number-theoretic methods.

Contribution

It establishes the non-integrality of $(k,l)$-G"obel sequences for a specific class of parameters, advancing understanding of their behavior.

Findings

Proves non-integrality for specific $(k,l)$ values.

Uses geometric arguments related to quadratic residues.

Contributes to unresolved questions about sequence integrality.

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.