FIBONACCI AND LUCAS NUMBERS AS PRODUCTS OF THEIR ARBITRARY TERMS - Daşdemir, Ahmet; Emin, Ahmet (Anadolu Universitesi Bilim ve Teknoloji Dergisi-A: Uygulamali Bilimler ve Muhendislik, 2024)
This study presents all solutions to the Diophantine equations F_k=L_m L_n and L_k=F_m F_n. To be clear, the Fibonacci numbers that are the product of two arbitrary Lucas numbers and the Lucas numbers that are the product of two arbitrary Fibonacci numbers are determined herein. The results under consideration are proven by using the Dujella-Pethő lemma in coordination with Matveev's theorem. All common terms of the Fibonacci and Lucas numbers are determined. Further, the Lucas-square Fibonacci and Fibonacci-square Lucas numbers are given.
Keywords: Temel Matematik (Diğer); Fibonacci and Lucas numbers; logarithmic height in logarithms; Matveev theorem; Dujella and Pethö lemma; Pure Mathematics (Other)