Non-linear Recurrences that Quite Unexpectedly Generate Rational Numbers

Details Non-linear Recurrences that Quite Unexpectedly Generate Rational Numbers Non-linear Recurrences that Quite Unexpectedly Generate Rational Numbers

2009-09-02

arxiv.org/abs/0909.0469v1

Mathematics

Combinatorics

14 pages, accompanying Maple code can be found at http://math.rutgers.edu/~eahogan/maple/maple.html

Scientific paper

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.

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