# A Pythagorean Introduction to Number Theory

## BY: Ramin Takloo-Bighash

## Announcement:

I am happy to announce that the book has been chosen as one of the “500 essential textbooks” that Springer is making freely available to help educators, students and academics affected by coronavirus lockdown. I'm very supportive of this initiative and I commend Springer for doing it. You can download the book for free from the Springer website.

## Book website and table of contents:

The table of contents can be found here.

## Errata:

Corrections and Suggestions, mostly by Dr. Mohammad Javaheri (Siena College) but also by Lior Silberman, Cheryl Plante, Alexa W.C. Lee-Hassan, Michalis Paizanis, and Andrés Gabriel Romero Arellano, can be found here.

Please send me an email at rtakloo@math.uic.edu if you find any mistakes.

## Sample exams and homework:

I will post a few sample exams here. If you teach a course from this book, I'd be happy to share your homework and exams here. For the spring 2020 course see below.

## Spring 2020:

Here I am posting the course materials from my spring 2020, Number Theory 2, course.

I started the course with Gaussian integers with the goal of proving the Two Square Theorem. Since some of the students had no seen any ring theory in their first semester abstract algebra course I covered the basic concepts of Euclidean domains, PIDs, and UFDs in a few lectures.

Here is a write up on Euclidean Domains, PIDs, and UFDs, and their relationships.

**Homework 1. **

Page 101: 5.1 (a, b, c), 5.2, 5.11

Page 253: A.3.3, A.3.5

Problem 1. Let S be the set of all polynomials with coefficients in Z that have no linear terms (i.e., the coefficient of x is 0). Show that S is a subring of Z[x]. Prove that x^6 can be written as a product of irreducible elements in two different ways. Can you determine the irreducible elements in S?

Problem 2. Prove that 2, 3, 1 + \sqrt{-5}, 1- \sqrt{-5} are irreducible in Z[\sqrt{5}]. Note that 6 = 2 x 3 = (1 + \sqrt{-5})(1 - \sqrt{-5}).

Problem 3. Show that if F is a field, F[x] is a Euclidean domain.

The next topic we covered in class was the Gauss circle problem from Chapter 9 followed by Geometry of Numbers from Chapter 10.

**Homework 2.**

Page 162: 9.2 (a, b, e, also do the other parts if you have taken complex analysis), 9.4

Page 180: 10.1, 10.11, 10.13, 10.16 (for this use SageMath, see the appendix), Challenge Problems: 10.12 , 10.4

The next topic covered was quaternions and their relationship to sums of squares from Chapter 11.

**Homework 3.**

Page 192: 11.2, 11.5, 11.8, 11.9

How many primitive right triangles are there with hypotenuse bounded by a large number B?

**Homework 4.**

Page 220: 13.3, 13.7, 13.8, 13.9, 13.14, 13.15

In Chapter 12 we discussed binary quadratic forms and we gave another proof for the Two Square Theorem. At the end I gave a brief presentation on the main ideas of the proof of the Three Squares Theorem, and Gauss's Composition Laws.

**Homework 5.**

Page 208: 12.3, 12.5, 12.6, 12.8, challenge: 12.14

The next topic we discussed was elliptic curves and congruent numbers.

**Homework 6.**

Page 72: 3.9, 3.19

Page 87: 4.10, 4.13

## Hints for Exercises:

I will be posting hints for the harder exercises here. If you find a problem in the book for which you need a hint, please send me an email.

## Questions or comments?

Please don't hesitate to reach out to me by emailing me at rtakloo@math.uic.edu.