Representation of and
Hùng Việt Chu · The Fibonacci Quarterly · 2020
Let a, b ∊ℕ be relatively prime. We consider (a – 1)(b – 1)/2, which arises in the study of the pqth 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 (Fn – 1)(Fn+1 – 1)/2 and (Fn – 1)(Fn+2 – 1)/2, where Fi is the ith Fibonacci number, and observe several nice patterns.