On the sum of three arbitrary Fibonacci and Lucas numbers

Abstract

In this paper, we solve the equations L-k = F-n + F-m + F-r,F- F-k = F-n + F-m + F-r, L-k = E-n + L-m + L-r, F-k = E-n + L-m + F-L, for 0 < r <= m <= n and a natural number k. It is shown that only the equation F-k = L-n+ L-m + L-r has a finite number of solutions. The others have infinitely many solutions.