#### 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.

#### Recommended Citation

Li, Huixi, "On Some Conjectures in Additive Number Theory" (2018). *All Dissertations*. 2209.

https://tigerprints.clemson.edu/all_dissertations/2209