Date of Award

8-2018

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

Committee Member

Dr. Jim Brown, Committee Chair

Committee Member

Dr. Michael Burr

Committee Member

Dr. Kevin James

Committee Member

Dr. Hui Xue

Abstract

In the first part of the thesis we prove that every sufficiently large odd integer can be written as a sum of a prime and 2 times a product of at most two distinct odd primes. Together with Chen's theorem and Ross's observation, we know every sufficiently large integer can be written as a sum of a prime and a square-free number with at most three prime divisors, which improves a theorem by Estermann that every sufficiently large integer can be written as a sum of a prime and a square-free number. In the second part of the thesis we prove some results that specialize to confirm some conjectures of Sun, which are related to Fermat's theorem on sums of two squares and other representations of primes in arithmetic progressions that can be represented by quadratic forms. The proof uses the equidistribution of primes in imaginary quadratic fields.

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