Diophantine equations with Lucas and Fibonacci number coefficients

Abstract

Fibonacci and Lucas numbers are special number sequences that have been the subject of many studies throughout history due to the relations they provide. The studies are continuing today, and findings about these number sequences are constantly increasing. The relations between the Fibonacci and Lucas numbers, which were found during the proof of the prime between two consecutive numbers belonging to the Fibonacci or Lucas number sequence with the Euclidean algorithm, started our project. In the project, Diophantine equations whose coefficients are Lucas or Fibonacci numbers have been studied, various relations have been found, and their proofs have been made. Fnx - Fn+1y = (-1)n, Lnx - Ln-1y = 1. As in the above example, the equivalents of x and y values were found in the Diophantine equations with Fibonacci and Lucas number coefficients; and based on this example, different variations of the Diophantine equations whose coefficients were selected from the Fibonacci and Lucas number sequences were created, and their proofs were made. Secondly, the geometric shapes consisting of vertices determined by pair of numbers selected from the Fibonacci or Lucas number sequence were considered, and their properties were examined. Various relations were found between them, and generalizations were made.