An integer-valued polynomial is a polynomial with rational coefficients that takes an integer value when evaluated at an integer. The binomial polynomials form a regular basis for the Z-module of all integer-valued polynomials. Using the idea of a p-ordering and a p-sequence, Bhargava describes a similar characterization for polynomials that are integer-valued on some subset of the integers. This thesis focuses on characterizing the polynomials that are integer-valued on the Fibonacci numbers.
For a certain class of primes p, we give a formula for the p-sequence of the Fibonacci numbers and an algorithm for finding a p-ordering using Coelho and Parry’s results on the distribution of the Fibonacci numbers modulo powers of primes. Knowing the p-sequence, we can then find a p-local regular basis for the polynomials that are integer-valued on the Fibonacci numbers using Bhargava’s methods. A regular basis can be constructed from p-local bases for all primes p.
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:NSHD.ca#10222/35316 |
| Date | 06 August 2013 |
| Creators | Scheibelhut, Kira |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English |
| Detected Language | English |
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