Thesis (Ph. D.)--University of Texas at Austin, 2005. / Supervisor: Fernando Rodriguez-Villegas. Vita. Includes bibliographical references.

[No abstract available] / Science, Faculty of / Mathematics, Department of / Graduate

[No abstract available] / Science, Faculty of / Mathematics, Department of / Graduate

The numbers 1, 2, and 6 have the same sum and same sum of squares as 0, 4, 5. These two sets are solutions of degree 2 of the Tarry-Escott problem. This problem of finding sets of integers having equal sums of like powers has been investigated for at least two hundred years and we have presented most of the general results. For any given k there exist solutions in integers of the system of equations (formula omitted) any solution will be composed of a set and a permutation of the set; such solutions are called trivial. Many writers have attempted to provide non-trivial solutions for the optimum case where s = k + 1. These so called ideal solutions exist for all k≦9 but no such solutions have been found for k≧10. We have been interested in providing solutions where s is smaller than for previous known examples, and have generated such solutions using a digital computer. Some of our results also apply to an extension of the Tarry-Escott problem in view of a result concerning bounds for this problem. / Science, Faculty of / Mathematics, Department of / Graduate

No description available.

In this thesis we prove several results in analytic number theory. 1. We show that there exist 3-digit palindromic primes in base b for a set of b having density 1 and that if b is sufficiently large then there is a $3$-digit palindrome in base b having precisely two prime factors. 2. We prove various estimates for averages of sums of Kloosterman fractions over primes. The first of these improves previous results of Fouvry-Shparlinski and Baker. 3. By using the q-analogue of van der Corput's method to estimate short Kloosterman sums we study the divisor function in an arithmetic progression to modulus q. We show that the expected asymptotic formula holds for a larger range of q than was previously known, provided that q has a certain factorisation. 4. Let ‖x‖ denote the distance from x to the nearest integer. We show that for any irrational α and any ϴ< 8/23 there are infinitely many n which are the product of two primes for which ‖nalpha‖ ≤ n ^-ϴ. 5. By establishing an improved level of distribution we study almost-primes of the form f(p,n) where f is an irreducible binary form over Z. 6. We show that for an irreducible cubic f ? Z[x] and a full norm form $mathbf N$ for a number field $K/Q$, satisfying certain hypotheses, the variety $$f(t)=mathbf N(x_1,ldots,x_k) e 0$$ satisfies the Hasse principle. Our proof uses sieve methods.

published_or_final_version / abstract / toc / Mathematics / Master / Master of Philosophy

published_or_final_version / Mathematics / Master / Master of Philosophy

No description available.

No description available.