# What topics to include on a first course in number theory

I need 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?

• Continued fractions and their application to music could be an interesting one (just as an example) – Sp3000 Jun 17 '13 at 16:31
• Induction, congruence classes/linear congruences, Chinese remainder theorem, Euler's Theorem, some sets/functions, groups/permutations, group theory (like langrange's theorem) – user67258 Jun 17 '13 at 16:34
• I find this question odd as usually courses at undergraduate level are determined according to the students' assumed interests and the departament's goals: are these mathematics students? Is the department strong in number theory? Has this first course to serve as a basis for continuation in algebraic number theory and/or close subjects or must it stand by itself and be only a "gentle" introduction to the subject...? – DonAntonio Jun 17 '13 at 16:59
• The goal is to highlight rigour but not to put off students. There is no continuation course and I think your description of a gentle introduction is a true reflection of what we want to achieve. – matqkks Jun 17 '13 at 21:41

You could do much worse than to use Underwood Dudley, Elementary Number Theory; it’s $35$ years old, but it’s very readable, it has all of the traditional topics, and it’s available in an inexpensive Dover paperback edition and an even more inexpensive Kindle edition.