Rational Polynomials That Take Integer Values at the Fibonacci Numbers

Abstract

An integer-valued polynomial on a subset S of ℤ is a polynomial f (x) ∊ ℚ [x] with the property f (S) ⊆ 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 nth one of degree n, with which any such polynomial can be expressed as a unique integer linear combination.

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Notes on contributors

Keith Johnson

KEITH JOHNSON received his Ph.D. in mathematics from Brandeis University and now teaches at Dal-housie University where he is a professor of mathematics. His interest in rings of integer-valued polynomials was originally sparked by their occurrence in algebraic topology.

Kira Scheibelhut

KIRA SCHEIBELHUT received her B.Sc. in mathematics from Dalhousie University in 2011. After a year spent traveling, she returned to Dalhousie and completed her M.Sc. in mathematics in 2013. She is currently working as an implementation analyst with the consulting firm Morneau Shepell while she contemplates yet another return to school.