Elementary Number Theory Syllabus: Teaching Tips & Resources

[matqkks]

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?

 

[lurflurf]

How many lectures are there? Do the students know any algebra? If not is teaching some discouraged? Some time on discrete math like sums, combinatorics, probability, and graph theory if they are needed for chosen topics. I think these are nice topics
Divisibility
-primes
-common functions
-Eucleidean algotithm
-chinese remainder
Continued fractions
-expanding numbers
-theorems
-surds
-Farey fractions
-periodicity
Congruence
-Fermat
-Wilson
-Linear
-multiple unknowns
-residue
-quadratic residue
-Jacobi symbol
-quadratic congruence
Diophantine equations
Algebraic numbers
-they are a ring
-a complex number rationally linear over some complex numbers is algebraic.
Estimating primes/prime number theorem
Partition
Density
Famous irrational numbers pi,e,phi...

As far as motivation I know offering to prove pi is irrational is very motivating for a minority of students. Many others will be glad to be learning something besides calculus. Others may be harder to move. There is a temptation to tempt students with the possibility of becoming rich and famous if they can figure out how to factor large numbers a million times faster than experts, but that is dishonest. Number theory has a number of application that can be mentioned, but the appeal is intellectual curiosity. Number theory is its own reward.