Investigations in continued fractions and covering sets

In 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.]