A DIOPHANTINE EQUATION INVOLVING C-NOMIAL COEFFICIENTS

Abstract

Let Cn be the nth Fibonacci number (Cn = Fn) or thenth Lucas number (Cn = Ln). For 1 ≤k ≤m, let m kC = CmCm−1 ···Cm−k+1 C1···Ck
be the corresponding C-nomial coefficient. In this paper, we prove that the only solutions of the Diophantine equation m kC = makb,
in positive integers m,k,a,b witha>1, are (m,k,a,b) = (1 ,1,a,b), (5,1,1,b), (12,1,2,b), and (5,3,1,1), for Cn = Fn and (m,k,a,b) = (1 ,1,a,b) in the case Cn = Ln.