Introduction to Number Theory
These are notes taken from Introduction to Number Theory - Berkeley Math 115 open lectures by fields medalist Prof. Richard Borcherds.
The textbook for the course is "An Introduction to the Theory of Numbers" by Niven, Zuckerman, and Montgomery. (5th edition)
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Introduction: In the first two lectures, a survey of some of the topics covered later in the course is given, mainly about primes and Diophantine equations. The survey continues with some more problems in number theory, and some discussion is made about congruences, quadratic reciprocity, additive number theory, recreational number theory, and partitions.
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Divisibility: This lecture covers basic properties of divisibility, and Euclid's algorithm for finding greatest common divisors. Then, we discuss how to solve linear equations in integers using Euclid's algorithm. Finally, we discuss some basic properties of primes and prove the fundamental theorem of arithmetic.
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Arithmetical Functions: Arithmetical functions are functions that are defined on positive integers; they are quite interesting. In the lecture, we give some examples of multiplicative functions and show how to calculate them. As an application we discuss even perfect numbers.
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Binomial Coefficients: The number of ways you can choose some elements from a set becomes something really cool real quick. In these two lectures, we review the definitions and basic properties of binomial coefficients. Then, we discuss some applications of binonial coefficients, such as an approximate estimate for the number of primes less than , and the Catalan numbers.
