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Using a classical result of Thue, we give an upper bound for the number of solutions to a family of quartic Thue equations. We also give an upper bound upon the number of solutions to a family of quartic Thue inequalities. Using the Thue-Siegel principle and the theory of linear forms in logarithms, an upper bound is given for general quartic Thue equations. As an application of the method of Thue-Siegel, we will resolve a conjecture of Walsh to the effect that the Diophantine equation aX⁴ - bY² = 1, for fixed positive integers a and b, possesses at most two solutions in positive integers X and Y. Since there are infinitely many pairs (a, b) for which two such solutions exist, this result is sharp. It is also effectively proved that for fixed positive integers a and b, there are at most two positive integer solutions to the quartic Diophantine equation aX⁴ - bY² = 2. We will also study cubic and quartic Thue equations by combining some classical methods from Diophantine analysis with modern geometric ideas. / Science, Faculty of / Mathematics, Department of / Graduate

This dissertation studies four problems in analytic and probabilistic number theory. Two of the problems are about a certain random number theoretic object, namely a random multiplicative function. The other two problems are about smooth numbers (i.e. numbers only having small prime factors), both in their own right and in their application to finding solutions to S-unit equations over the integers. Thus all four problems are concerned, in different ways, with _understanding the multiplicative structure of the integers. More precisely, we will establish that certain sums of a random multiplicative function satisfy a normal approximation (i.e. a central limit theorem) , but that the complete sum over all integers less than x does not satisfy such an approximation. This reflects certain facts about the number and size of the prime factors of a typical integer. Our proofs use martingale methods, as well as a conditioning argument special to this problem. Next, we will prove an almost sure omega result for the sum of a random multiplicative function, substantially improving the existing result of Halasz. We will do this using a connection between sums of a random multiplicative function and a certain random trigonometric sum process, so that the heart of our work is proving precise results about the suprema of a class of Gaussian random processes. Switching to the study of smooth numbers, we will establish an equidistribution result for the y-smooth numbers less than x among arithmetic progressions to modulus q, asymptotically as (logx)/(logq)-+ oo, subject to a certain condition on the relative sizes of y and q. The main point of this work is that it does not require any restrictions on the relative sizes of x and y. Our proofs use a simple majorant principle for trigonometric sums, together with general tools such as a smoothed explicit formula. Finally, we will prove lower bounds for the possible number of solutions of some S-unit equations over the integers. For example, we will show that there exist arbitrarily large sets S of prime numbers such that the equation a+ l = c has at least exp{(#S)116- �} solutions (a, c) with all their prime factors from S. We will do this by using discrete forms of the circle method, and the multiplicative large sieve, to count the solutions of certain auxiliary linear equations.

Three concepts of ordinal numbers are examined with a view to their intuitiveriess and existence in two principle systems of axiomatic set theory. The first is based on equivalence classes of the similarity relation between well-ordered sets. Two alternatives are suggested in later chapters for overcoming the problems arizing from this definition. Next, ordinal numbers are defined as certain representatives of these equivalence classes,, and one of several such possible definitions is taken for proving the fundamental properties of these ordinals. Finally, a generalization of Peano's axioms provides us with a method of defining ordinal numbers which are the ultimate result of abstractions. / Science, Faculty of / Mathematics, Department of / Graduate

In Monte Carlo calculations performed on electronic computers it is advantageous to use an arithmetic scheme to generate sets of numbers with "approximately" the properties of a random sequence. For many applications the local characteristics of the resulting sequence are of interest. In this thesis the concept of a pseudo-random sequence is set out, and arithmetic methods for their generation are discussed. A brief survey of some standard statistical tests of randomness is offered, and the results of empirical tests for local randomness performed on the ALWAC III-E computer at the University of British Columbia are recorded. It is demonstrated that many of the standard generating schemes do not yield sequences with suitable local properties, and could therefore be responsible for misleading results in some applications. A method appropriate for the generation of short blocks of numbers with approximately the properties of a randomly selected set is proposed and tested, with satisfactory results. / Science, Faculty of / Mathematics, Department of / Graduate

This thesis continues the studies of A. Waksman (1969) in the repeated generation of finite strings in a one dimensional array of finite automata. Waksman handles this problem by the use of a "modulo arithmetic" algorithm. This is shown to be very restrictive with regard to the number of characters permitted in the output string. In fact, it is shown that unless the length of the string which is to be repeated is of the form p(formula omitted) where p is prime, only one output character is permitted. This of course makes the process quite meaningless. For this reason, a new algorithm is developed. This is referred to as the wheel algorithm, since there is an obvious analogy between it and a wheel, with the output string on its circumference rolling along the array and leaving the imprint of the characters in the string behind it in the same way that a wheel leaves tire tracks. The number of states required for such an algorithm is large and so the binary wheel algorithm is introduced. By using this algorithm, in which an output state is represented as a string of bits in several cells, the number of states required, in addition to the n output states, can be reduced to about 4 log₂ n. Both the wheel and binary wheel algorithms are then extended to the two dimensional and finally the d-dimensional cases. / Science, Faculty of / Computer Science, Department of / Graduate

This thesis deals with the properties of the coefficients of Hypergeometric Series. Specifically, we are interested in which primes appear in the denominators to a bounded power. The first main result gives a method of categorizing the primes up to equivalence class which appear finitely many times in the denominators of generalized hypergeometric series nFm over the rational numbers. Necessary and sufficient conditions for when the density is zero are provided as well as a categorization of the n and m for which the problem is interesting. The second main result is a similar condition for the appearance of primes in the denominators of the hypergeometric series 2F1 over number fields, specifically quadratic extensions Q(D). A novel conjecture to the study of p-adic numbers is also provided, which discusses the digits of irrational algebraic numbers' p-adic expansions. / Thesis / Master of Science (MSc)

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