A DIOPHANTINE EQUATION INVOLVING C-NOMIAL COEFFICIENTS

Abstract

Let Cn be the nth Fibonacci number (Cn = Fn) or the nth Lucas

number (Cn = Ln). For 1 ≤ k ≤ m, let

m k C = CmCm

−

1

• ··

C

m−k+1

C

1

• ··

C

k

be the corresponding C-nomial coefficient. In this paper, we prove that

the only solutions of the Diophantine equation

m k C = makb ,

in positive integers m, k, a, b with a > 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.