Some integer sequences and Pell equations


Thesis Type: Doctorate

Institution Of The Thesis: Uludağ Üniversitesi, Turkey

Approval Date: 2013

Thesis Language: Turkish

Student: ARZU ÖZKOÇ

Supervisor: AHMET TEKCAN

Abstract:

In this work, some algebraic properties are obtained on integer sequences and Pell equations.In the first section, the preliminary notations, definitions and theorems which are to be used in later sections are given.In the second section, two separate integer sequences with two and four parameters are discussed and some algebraic relations on them are derived. Moreover, a Pell equation is defined using the parameters of the sequence with two parameters, and deduced the integer solutions of it including some recurrence relations. IN the third section, oblong numbers, Pell form and Pell equations have been considered. The integer solutions of Pell equation can be deduced by using oblong numbers. Also the cycle and proper cycle of the reduction of Pell form are given. Moreover, it is proved that the set of proper automorphisms of Pell form can be obtained by using the matrices related to oblong numbers. In the fourth section balancing numbers are considered. Some algebraic relations on them and also their relationships with Pell and Pell-Lucas numbers are deduced. Further, some formulas on perfect squares and Pythagorean triples related to balancing numbers are given. In the last section, integer solutions of Diophantine equation x^2-(t^2-t)y^2-(4t-2)x+(4t^2-4t)y=0 for an integer t>=2 is considered over Z and also over finite fields F_p for primes p>=5.