Mathematics Senior Capstone Papers

Document Type

Article

Publication Date

Spring 2024

Abstract

This paper serves as an extension/application of a method detailed by Chang et. al. in two separate papers which worked with sums modulo p for combinatorial sequences, positive integers r, and prime numbers p. Using a known formula, the Fibonacci numbers were converted into a sum of binomial coefficients to apply the aforementioned method. A hypothesis was formed using computational methods, where a pattern was observed to hold for the first 50,000 prime numbers. This claim was then proven following the methodology from Chang et. al. using properties of Laurent polynomials, definitions and theorems related to the Fibonacci numbers, and elements of modular arithmetic including the freshman’s dream congruence identity and quadratic residues. It was found that for r = 1, the sum is congruent to -1 mod p if p is congruent to 2 or 3 mod 5, to 0 mod p if p is congruent to 1 or 4 mod 5, and to 2 mod p if p is 5. For r = 2, the sum is congruent to 0 mod p if p is congruent to 2 or 3 mod 5, to 1 mod p if p is congruent to 1 or 4 mod 5, and to 3 mod p if p is 5. The paper concludes with a conjecture regarding the sum of the first rp Fibonacci numbers for arbitrary r.

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