What to include in an introduction?

[matqkks]

I am writing an introduction to a first course in elementary number theory. The topics are linear Diophantine equations, modular arithmetic including FLT and Euler's Generalization, quadratic residues and Non - linear Diophantine equations.
How can I write an introduction to this showing linkage between the various topics and hook potential students to do this course? What is the motivation on covering these topics?

 

[Wilmer]

I am writing an introduction to a first course in elementary number theory.
The topics are linear Diophantine equations, modular arithmetic including FLT
and Euler's Generalization, quadratic residues and Non - linear Diophantine equations.
How can I write an introduction to this showing linkage between the various topics
and hook potential students to do this course?
What is the motivation on covering these topics?

Hmmm...perhaps what "hooked" YOU...get my drift?

 

[Petek]

Possible topics that may fit your requirements:
1. Fermat's Last Theorem. Original statement by Fermat as a marginal note. Margins to small to contain Fermat's alleged proof. Attemps to solve led to advances in other ares of math. Large prize offered for a solution, leading to many "crackpot" solutions. Ultimate proof used math unknown in Fermat's time.

2. Number theory and cryptography. Number theory provides computationally complex problems (e.g., prime factorization, elliptic curve logarithm problem) that lead to codes that are unbreakable in practice. Used in credit card encryption and other situations.

3. Archimedes's cattle problem (see here). Leads to a non-linear Diophantine equation (i.e., Pell's Equation). The solution has more than 200 000 digits and wasn't written out until the advent of digital computers.