Representation of (1)/(2)(F_n-1)(F_n+1-1) and (1)/(2)(F_n-1)(F_n+2-1)

2020/01/15 by Hùng Việt Chu, Chu, Hung Viet
Mathematics · #11B39 #Advanced Algebra and Geometry #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2001.08066

openalex publication_date 2020/01/15 · openalex created_date 2020/01/30 · openalex updated_date 2026/07/28

Abstract

Let a, b∈ ℕ be relatively prime. We consider (a-1)(b-1)/2, which arises in the study of the pq-th cyclotomic polynomial, where p,q are distinct primes. We prove two possible representations of (a-1)(b-1)/2 as nonnegative, integral linear combinations of a and b. Surprisingly, for each pair (a,b), only one of the two representations exists and the representation is also unique. We then investigate the representations of (F_n-1)(F_n+1-1)/2 and (F_n-1)(F_n+2-1)/2, where F_i is the i^th Fibonacci number, and observe several nice patterns.

Citations

• Fibonacci Numbers and Identities II
• The Midterm Coefficient of the Cyclotomic Polynomial F pq (x)

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