"Mathematics is the queen of the sciences and number theory is the queen of mathematics." Carl Friedrich Gauss
Number theory, the study of the integers, is a vibrant area of mathematical research that many students do not have the opportunity to study in high school. The objectives for this course are to expose students to this beautiful theory, to understand what inspired this quote from Gauss, and to allow students to experience mathematics as a creative, empirical science.
We will begin by studying the axioms of the integers, and spend the next three weeks unraveling their properties: divisibility, modular arithmetic, solving linear Diophantine equations, the Chinese remainder theorem, prime factorization, the Gaussian integers, and quadratic reciprocity, to name a few. While exploring these concepts, students will learn how to write rigorous proofs of the theorems they discover. Much time will be devoted to learning proof techniques and logic. We will also discuss applications of number theory to cryptography.
When one studies mathematics in college, there is a transition that takes place after calculus from courses focused on solving problems to courses focused on proving theorems. In this class, we will take the latter approach to give students an idea of what college-level math is like. Lecture time will be minimized, as students will be encouraged to discover properties of the integers rather than being told what they are. Via small group work, students will observe patterns, make conjectures, and try to prove them. At the end of the course, students will know what it’s like to think and work like a mathematician.
By the end of this course, students will be able to write a rigorous mathematical proof and will have developed a repertoire of proof techniques. They will also know quite a bit of elementary number theory, including the law of quadratic reciprocity, how to solve linear Diophantine equations, how to use modular arithmetic, and how to encode and decode messages with public-key cryptography.