Non-linear Recurrences that Quite Unexpectedly Generate Rational Numbers

TL;DR

This paper explores non-linear recurrences, especially those that are not linear in the highest order term, revealing surprising cases where they generate rational numbers infinitely, contrary to typical expectations.

Contribution

It introduces new examples of non-linear recurrences that produce rational numbers infinitely, expanding understanding beyond linear recurrence behaviors.

Findings

Certain non-linear recurrences generate rational numbers ad infinitum.

Examples include a first-order recurrence with m=2 and a third-order recurrence from a generalized Somos sequence.

The behavior differs significantly from traditional linear recurrence results.

Abstract

Non-linear recurrences which generate integers in a surprising way have been studied by many people. Typically people study recurrences that are linear in the highest order term. In this paper I consider what happens when the recurrence is not linear in the highest order term. In this case we no longer produce a unique sequence, but we sometimes have surprising results. If the highest order term is raised to the $m^{th}$ power we expect answers to have $m^{th}$ roots, but for some specific recurrences it happens that we generate rational numbers ad infinitum. I will give a general example in the case of a first order recurrence with $m=2$, and a more specific example that is order 3 with $m=2$ which comes from a generalized Somos recurrence.