Interesting Esoterica

Rational Polynomials That Take Integer Values at the Fibonacci Numbers

Article by Keith Johnson and Kira Scheibelhut
  • Published in 2016
  • Added on
An integer-valued polynomial on a subset $S$ of $\mathbb{Z}$ is a polynomial $f(x) \in \mathbb{Q}[x]$ with the property $f(S) \subseteq \mathbb{Z}$. This article describes the ring of such polynomials in the special case that $S$ is the Fibonacci numbers. An algorithm is described for finding a regular basis, i.e., an ordered sequence of polynomials, the $n$th one of degree $n$, with which any such polynomial can be expressed as a unique integer linear combination.

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BibTeX entry

@article{RationalPolynomialsThatTakeIntegerValuesattheFibonacciNumbers,
	title = {Rational Polynomials That Take Integer Values at the Fibonacci Numbers},
	author = {Keith Johnson and Kira Scheibelhut},
	url = {http://www.jstor.org/stable/10.4169/amer.math.monthly.123.4.338},
	abstract = {An integer-valued polynomial on a subset {\$}S{\$} of {\$}\mathbb{\{}Z{\}}{\$} is a polynomial {\$}f(x) \in \mathbb{\{}Q{\}}[x]{\$} with the property {\$}f(S) \subseteq \mathbb{\{}Z{\}}{\$}. This article describes the ring of such polynomials in the special case that {\$}S{\$} is the Fibonacci numbers. An algorithm is described for finding a regular basis, i.e., an ordered sequence of polynomials, the {\$}n{\$}th one of degree {\$}n{\$}, with which any such polynomial can be expressed as a unique integer linear combination.},
	comment = {},
	urldate = {2016-08-02},
	year = 2016
}