Research Information System
Mathematica Slovaca, vol.74, no.3, pp.563-576, 2024 (SCI-Expanded)
Let (Fn)n≥0 and (Ln)n≥0 be the Fibonacci and Lucas sequences, respectively. In this paper we determine all Fibonacci numbers which are mixed concatenations of a Fibonacci and a Lucas numbers. By mixed concatenations of a and b, we mean the both concatenations ab and ba together, where a and b are any two nonnegative integers. So, the mathematical formulation of this problem leads us searching the solutions of two Diophantine equations Fn = 10d Fm + Lk and Fn = 10d Lm + Fk in nonnegative integers (n, m, k), where d denotes the number of digits of Lk and Fk, respectively. We use lower bounds for linear forms in logarithms and reduction method in Diophantine approximation to get the results.