# Cluster 1: Number Theory and Discrete Math

(This is a FIRST CHOICE option only)

**Prerequisite:**One year of high school mathematics. Two years preferred. Algebra 1 minimum.

All students in this cluster will be enrolled in the following courses:

### Number Theory

#### Instructor: Richard Gottesman, PhD Candidate

Number theory is one of the oldest areas of mathematics and is also one of the most active fields of current mathematical research.

Many problems in number theory are easy to state but making progress towards solving them often requires creative insights. An example is the problem of determining whether there are infinitely many pairs of primes whose difference is 2. This problem is still unsolved but much progress has been made in recent years.

In this course, we will learn how to prove mathematical theorems. We will also focus on experimenting with numbers, coming up with conjectures, and hopefully proving our conjectures. It will be a lot of fun! Some of the topics we will explore include the uniqueness of prime factorization, modular arithmetic, mathematical induction, Fibonacci numbers, and prime numbers. We will also focus on determining which positive integers are sums of two squares. One of the highlights of this course is the quadratic reciprocity law.

### Discrete Math: Infinity, Graph Theory, and Cryptography

#### Instructor: Joseph Ferrara, PhD Candidate

This course will serve as an introduction to three topics, highlighting different ways of thinking and doing mathematics. The first topic is infinity, where the notions of sets and functions will be introduced. Infinity, being a difficult concept to fully grasp gives a taste of abstraction in mathematics, and the discussion of sets introduces language that will be used the rest of the course. After infinity, we will come down to Earth and learn some graph theory, beginning with the famous problem from the 1700s of the seven bridges of Konigsberg. Graph theory has a more geometric approach and flavor, being a subject that one can literally see. Finally the last part will be an introduction to cryptography. Using the concepts learned from the number theory course (which is the other course offered in this cluster), an introduction to public key cryptography will be given, including a discussion of the RSA algorithm. This part of the course has an algorithmic and real world applications feeling to it.

Along with introducing the topics mentioned, many mathematical problems and puzzles will be given throughout the course. Problem solving in mathematics is very important, and we will spend time in class solving and presenting interesting problems. As a famous mathematician, Paul Halmos, once said, "The only way to learn mathematics, is to do mathematics!"

### Transferable Skills: Tools for Success

It may or may not surprise you that being a university researcher requires a whole host of skills outside of the specific scientific knowledge required of your chosen discipline or specialty. It requires communication skills such as the ability to present your work in writing and orally. It requires competencies in the realm of information technology including the ability to find and judge (the validity of) information and use a variety of hardware and software tools (e.g. spreadsheets, databases, statistics software, other data manipulation tools). It requires all of those skills to effectively conduct research such as data collection, analysis and interpretation, critical thinking and problem solving as well as the ability to conduct laboratory and/or field work. And, of course, a baseline competency in English, science, mathematics and computers is critical.

The governing mission of the UCSC COSMOS Transferable Skills course is to promote students’ future academic (and professional) success through the exploration and development of transferable skills: i.e. those competencies that students develop while in school which facilitate academic achievement, the eventual transition into the work-force and which are applicable in many other life situations.