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

 dc.contributor.author Mabaso, Automan Sibusiso dc.date.accessioned 2021-12-13T16:53:56Z dc.date.available 2021-12-13T16:53:56Z dc.date.issued 2021 dc.description 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 en_ZA dc.description.abstract 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. en_ZA dc.description.librarian TL (2021) en_ZA dc.faculty Faculty of Science en_ZA dc.identifier.uri https://hdl.handle.net/10539/32300 dc.language.iso en en_ZA dc.phd.title PHD en_ZA dc.school School of Mathematics en_ZA dc.title Solving some diophantine equations involving fibonacci numbers, catalan numbers, ramanujan function and factorials en_ZA dc.type Thesis en_ZA
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