Understand the logic of mathematical theory. Figuring out how Gauss contributed to number theory mandates understanding mathematical thinking. Specifically, mathematicians use deductive reasoning to generate "theorems," which are then proved by writing "proofs." Proofs demonstrate through assumptions and justified conclusions the validity of the theorem.
Define number theory. Number theory is the study of natural numbers, which are defined as numbers used in everyday life, such as 0, 1, 2, 3 and 4. Research on natural numbers is what is generally understood as algebra, such as division, multiplication and comparison of the behaviors and qualities of numbers. Cryptology is one example of applied number theory.
Read about the basics of number theory. Number theory encompasses composite numbers, factors, prime numbers and patterns in multiplication. Mathematicians from Euclid to Gauss have focused on defining and questioning the behaviors and patterns in prime numbering, for instance.
Read a summary of the theories that lay the foundation for Gauss's work on numbers. In his "Number Research" book, Gauss summarizes and builds upon theories written by mathematicians Euler, la Grange and Legendre.
Think in the abstract. Mathematicians who study number theory use specialized language and terminology to pose theoretical questions. For instance, typical language can sound something like this: "We will want to use the language of ideals, but we will not assume any prior familiarity with them. Suppose n is an integer, and consider the set (n) of multiples of n," a sentence in an article by Henry Cohn, Joshua Greene and Jonathan Hanke on algebraic equations, which expand on Gauss's works.
Recognize that while you can understand the basics of number theory, you may not be able to wholly figure out all of Gauss's works. This mathematician spent a lifetime grappling with numbers. Gauss is credited with saying that "if others would but reflect on mathematical truths as deeply and continuously as I have, they would make my discoveries." It is possible these theorems are universal truths waiting to be discovered, as he posits; however, higher level comprehension of his number theories requires aptitude and advanced study.