Diophantine Equation in Logarithms

Abstract

The main work of these pages is written by myself under the supervisor of Dr. Omar Kihel, pertaining to continued fractions and applications , linear form in logarithms and the solutions of Diophantine equation Fn1 + Fn2 + Fn3 + Fn4 = 6a . The initial aim of the paper was to explore the possible solutions of the Diophantine equations in the form of Fn1 +Fn2 +Fn3 +Fn4 = y a . I begin my thesis by establishing some preliminary results and applications. The paper managed to extend the ideas of results of the Diophantine equations Fn1 +Fn2 +Fn3 +Fn4 = 2a and Fn1 +Fn2 +Fn3 +Fn4 = 11a . Mattveev Theorem, Legendre Theorem and a lemma by Dujella-petho are key theorems which we establish the main result. This paper includes the result of Diophantine equation Fn1 +Fn2 +Fn3 +Fn4 = 6a and it may require computations by computers. I will begin by introducing continued fractions, leading to linear forms in logarithms, followed by a section on the necessary preliminaries on Fibonacci numbers which concludes my results of the sum of four Fibonacci numbers. I then move to explore the aforementioned solutions of Fn1 + Fn2 + Fn3 + Fn4 = 6a .