Thesis-2004-Nag.pdf (1021.08 kB)

Download file# Investigations in continued fractions and covering sets

thesis

posted on 2018-08-06, 11:16 authored by Robin NagIn the first chapter we investigate matters regarding the period of continued
fractions of real numbers of the form √N = √(d² + r), where 1 ≤ r ≤ 2d. We also
derive an algorithm that can be used to generate partial quotients of continued
fractions of this type. We obtain a bound for the average period of continued
fraction expansions for fixed values of d. Finally, we obtain asymptotic
approximation formulae which estimate the number of N ≤ x such that the period
of the continued fraction expansion for √N is a fixed positive integer value.
In the second chapter, our objective is to express the set of all positive integers
as a finite collection of ai (mod mi), 1 ≤ i ≤ k, where k is a sufficiently
large integer, such that the moduli mi are distinct and mi ≥ 8.
In order to do this, we must show that for any given integer n, it is easy to
verify that n ≡ ai (mod mi) for some i in 1 ≤ i ≤ k.
We shall show that there is a proof that the union of ai (mod mi), 1 ≤ i ≤ k
covers (i.e. includes) the set of all integers. In order to do this we shall use
a method given by R. Morikawa in [5] to construct a covering congruence tree
which contains the necessary collection of ai (mod mi).
In the third chapter, we prove that it is impossible to cover the set of all positive
integers as a finite collection of ai (mod mi), 1 ≤ i ≤ k, where k is a sufficiently
large integer, such that the moduli mi are distinct, co-prime, odd and greater
than one. Furthermore, we prove that there exists an infinite set of arithmetic
progressions of integers which are not covered. We note that this was previously
an unsolved problem on which no significant progress had been made. [Continues.]

## History

## School

- Science

## Department

- Mathematical Sciences