Anticipated Date of Graduation

Spring 2024

Document Type

Thesis

Degree Name

Master of Science in Mathematical Sciences

Department

Mathematical Sciences

First Advisor

Doug Darbro

Abstract

The thesis focuses on a relatively recent theorem (2017) contributed by Adrian Dudek to the field of Number Theory. It states that every integer greater than two can be represented as a sum of a prime and a square-free number. This result immediately attracts attention since it bears similarities with the famous and not yet proved Goldbach’s Conjecture, namely that every even number larger than two can be represented as a sum of two primes. The expository analysis aims to understand the theorem in the general setting of the Additive Number Theory, the method of proof and previous results on which it stands, its significance and implications. We show that while square-free numbers are significantly more abundant among integers than prime numbers, they have some significant properties of primes. In this sense, the result can be related to Goldbach’s conjecture. To some extent, a theory similar to the theory of primes is developed for square-free numbers as well, e.g., counting functions, asymptotic density, and explicit estimates for the average density. Number Theory contains numerous theorems that are easy to state us-ing familiar concepts in Arithmetic. However, the proofs of these theorems are often complex, requiring tools from other areas of Mathematics such as Real Analysis, Complex Analysis, Approximation Theory, Numerical Analy-sis, and Computer Simulations. This is precisely the case with the proof of this theorem. It involves a theoretical part, where prior results linked to the mentioned areas are very cleverly combined to obtain the theorem for num-bers larger than 1010. For smaller numbers, the proof is based on numerical computations. In the dissertation, we discuss the previous findings, their role in logical deductions, and the unique way they fit together to yield the proof.

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