Generalized Fibonacci and Lucas sequences and their appications

The aim of this study is to observe the sequences U_n , V_n and u_n , v_n which are the generalizations of the Fibonacci and Lucas sequences and to investigate all solutions of some Diophantine equations, by using properties of these sequences.In the first section, definitions of the Fibonacci and Lucas sequences are given. Furthermore, the divisibility properties of the elements of these number sequences are proved.In the second section, generalizations of the Fibonacci and Lucas sequences are defined. Some new identities related to the generalized Fibonacci and Lucas sequences are proved. Furthermore, relations between the generalized Fibonacci and Lucas sequences and all integer solutions of some Diophantine equations are characterized.In the third section, the divisibility properties of the sequences U_n , V_n and u_n , v_n are proved.Finally, in the fourth section, by using the matrices which have the elements of the generalized Fibonacci and Lucas numbers, some identities are obtained. By considering these identities, all integer solutions of different Diophantine equations are obtained in some theorems.