# Number TheoryWhat to include on a first elementary number theory course?

#### matqkks

##### Member
I have to teach a course in elementary number theory next academic year. What topics should be included on a first course in this area? What is best order of doing things? The students have a minimum background in proof but this is a second year undergraduate module.
I am looking for applications which will motivate the student in this subject.
Are there good resources on elementary number theory?

MHB Math Helper

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
I like chapter 4 "Rational Approximations" from the book "Roots to Research: A Vertical Development of Mathematical Problems" by J.D. Sally and P.J. Sally, Jr. It can be downloaded from the AMS site by clicking on the "Preview Material". The book description says that it is for students of all levels starting from high school. I myself studied this material as an undergraduate, but also in a university-preparatory school.

Another interesting topic that touches number theory is squaring the circle. It took us a third of a semester during the fourth year in college to cover the theorem that $\pi$ is transcendental, so this fact should probably be given without a proof. However, it is interesting why every line segment built from a segment of length 1 using only compass and straightedge has an algebraic length. I don't think I have ever studied a rigorous proof. However, this may be more of a geometrical than number-theoretical theorem.

Finally, I would look at public-key cryptography, but I don't know if it is possible to omit enough details to make it into both an accessible and a rigorous course.

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
Perhaps the the following book by Graham Everest and Thomas Ward will provide you good ideas

An Introduction to Number Theory
From the title page:

"An Introduction to
Number Theory

With 16 Figures"

I wanted to exclaim, "Yeah, this is not geometry, baby! You should be happy with just a handful of pictures". Also, the beginning of "Alice in Wonderland" comes to mind.

Alice was beginning to get very tired of sitting by her sister on the bank, and of having nothing to do: once or twice she had peeped into the book her sister was reading, but it had no pictures or conversations in it, and what is the use of a book,' thought Alice without pictures or conversation?'

#### Bacterius

##### Well-known member
MHB Math Helper
Finally, I would look at public-key cryptography, but I don't know if it is possible to omit enough details to make it into both an accessible and a rigorous course.
I don't think so. It's fun and all as an example of applications of number theory but without computational complexity theory and an actual course in cryptography the students are going to be lost. Cryptography needs its own separate course IMHO

Also, on topic: modular arithmetic. The single most important tool. No elementary number theory course is complete without it.