Fibonacci numbers which are concatenation of three Fibonacci or Lucas numbers

Abstract

In this note, we show that there is no Fibonacci number that can be expressed as a concatenation of three Fibonacci or Lucas numbers under a certain constraint. That is, we solve to the Diophantine equations

F_n = 10^d+lF_m1 + 10^lF_m2 + F_m3

and

F_n = 10^d+lL_m1 + 10^lL_m2 + L_m3

in non-negative integers (n, m1, m2, m3) with m2 ≤ m1, where d and l represent the number of digits of the (Fm2, Lm2) and (Fm3, Lm3), respectively. Moreover, we give these equations as a problem to the researchers without the constraint m2 ≤ m1.