Chapter 1 - Divisibility and Factorization

The first chapter of the book introduces you to the basics of number theory, getting you comfortable with playing around with integers, introducing you to primes, and asking some basic questions about divisors and multiples. We'll cover roughly one section per day. The notes for each course will be posted as they become available.

- Lecture 0 - An Introduction to Numbers
- We started today by getting to know the policies and expectations in the course. All of this is available already on the syllabus, but if you have any questions don't be shy about moc.liamg|ztluhcs.c.werdna#ydnA gniliame. We also spent some time introducing ourselves briefly; this will be continued as you post your own profiles for Homework 0. Afterwards, we started talking about the basics in number theory, starting with the axioms. We finished by introducing the notion of divisibility for the integers.
- Lecture 1 - Introducing Divisibility
- Today we continued our discussion of divisibility and its basic properties. We saw some examples of how to put these properties into practice to prove exciting new results which might otherwise be quite difficult. Today's lecture culminated in the statement and proof of the division algorithm, one of the foundational results in number theory.
- Lecture 2 - Greatest Common Divisors
- Last class period we talked at length about divisibility and the division algorithm. Today we moved on to discuss the concept of greatest common divisors. We finished by describing some properties that greatest common divisors enjoy, including the (surprisingly powerful) result that the gcd of two integers
*a*and*b*can be expressed as an integral linear combination of*a*and*b*. - Lecture 3 - The Euclidean Algorithm; Prime Numbers
- Although we introduced the concept of greatest common divisors in class last time, we didn't come up with a very effective way of computing GCDs. We remedy this with the Euclidean Algorithm, and we show how this algorithm can also be used to express the GCD of two numbers
*a*and*b*as an explicit linear combination of*a*and*b*. Afterwards we introduced prime numbers and started proving some results about them. - Lecture 4 - Prime Numbers; the Fundamental Theorem of Arithmetic
- Today we spent the first half of the class exploring questions about prime numbers. Along the way we proved that there are infinitely many prime numbers and that there are arbitrarily large gaps between prime numbers. We also saw a formula which gives a rough count for the number of integers up to a given number
*x*, and we saw some conjectures about other behaviors about prime numbers. In the last half of the class we started a proof of the Fundamental Theorem of Arithmetic. - Lecture 5 - The Fundamental Theorem and its Applications
- Today we began by finishing off the proof of the fundamental theorem of arithmetic. After we completed the proof, we saw how the fundamental theorem could be used to facilitate the computation of GCDs and LCMs, and we also used it to prove that there are infinitely many primes which leave remainder 3 after division by 4. Finally, we got a sneak peak at the fundamental concept in chapter 2: congruence of integers.