A thesis submitted to the Faculty of Science, University of the
Witwatersrand and to the University Cheikh Anta Diop of Dakar(UCAD)
in fulfillment of the requirements for a Dual-degree for Doctor in
Philosophy in Mathematics. November 6th, 2017. / This thesis is about the study of Diophantine equations involving binary recurrent
sequences with arithmetic functions. Various Diophantine problems are investigated
and new results are found out of this study. Firstly, we study several
questions concerning the intersection between two classes of non-degenerate binary
recurrence sequences and provide, whenever possible, effective bounds on
the largest member of this intersection. Our main study concerns Diophantine
equations of the form '(jaunj) = jbvmj; where ' is the Euler totient function,
fungn 0 and fvmgm 0 are two non-degenerate binary recurrence sequences and
a; b some positive integers. More precisely, we study problems involving members
of the recurrent sequences being rep-digits, Lehmer numbers, whose Euler’s
function remain in the same sequence. We prove that there is no Lehmer number
neither in the Lucas sequence fLngn 0 nor in the Pell sequence fPngn 0. The
main tools used in this thesis are lower bounds for linear forms in logarithms
of algebraic numbers, the so-called Baker-Davenport reduction method, continued
fractions, elementary estimates from the theory of prime numbers and sieve
methods. / LG2018
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:wits/oai:wiredspace.wits.ac.za:10539/24996 |
| Date | January 2017 |
| Creators | Faye, Bernadette |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | Online resource (84 leaves), application/pdf |
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