Solving some diophantine equations involving fibonacci numbers, catalan numbers, ramanujan function and factorials

Mabaso, Automan Sibusiso
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In this thesis we study some Diophantine equations involving Fibonacci numbers, Catalan numbers, Ramanujan τ -function and Factorials. Since there is no generic method or algorithm that can be used in solving all Diophantine equations, the arithmetic properties of Ramanujan τ -function, Catalan numbers and Fibonacci numbers will play an important role. For some Diophantine equations, we will compare the order at which some small prime, say 2, divides the left and right-hand side of the equation. In some cases, we will use lower bound for nonzero linear forms in logarithms due to Laurent Mignotte and Nesterenko. Firstly, we solve some Diophantine equations of the form |τ (x)| = y, where τ is the Ramanujan τ–function and x, y are integer variables restricted to values of factorials, Fibonacci numbers and Catalan numbers. Our study in this thesis also includes an analysis of the Diophantine equation of the form Fn = ±τ (m1!) ± · · · ± τ (mk!), where Fn is the nth Fibonacci number and τ is the Ramanujan τ–function. We find some bounds for k, mk and show that when k = 2, the only positive integer solution of the Diophantine equation Fn = ±τ (m1!) ± τ (m2!), where m1 ≤ m2 is (1, 1, 3). Lastly, we do an analysis on the iterates of the Ramanujan τ–function and come up with some lemmas and propositions with respect to greatest prime factors and counting the number of solutions of some equations involving them.
A dissertation submitted in fulfilment of the requirements for the degree of Doctor of Philosophy (Mathematics) to the Faculty of Science, School of Mathematics, University of the Witwatersrand, Johannesburg, 2021