2024/08/21 by Xuyuan Chen, Chen, Xuyuan, Hùng Việt Chu +13 · 2 citations
Physics and Astronomy · Mathematics · #11B39 #11D04 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2409.02933
Let a, b∈ ℕ be relatively prime. Previous work showed that exactly one of the two equations ax + by = (a-1)(b-1)/2 and ax + by + 1 = (a-1)(b-1)/2 has a nonnegative, integral solution; furthermore, the solution is unique. Let Fn be the nth Fibonacci number. When (a,b) = (Fn, Fn+1), it is known that there is an explicit formula for the unique solution (x,y). We establish formulas to compute the solution when (a,b) = (Fn2, Fn+12) and (Fn3, Fn+13), giving rise to some intriguing identities involving Fibonacci numbers. Additionally, we construct a different pair of equations that admits a unique positive (instead of nonnegative), integral solution.