What is the most motivating way to introduce primitive roots?

[matqkks]

I am teaching elementary number theory to first year undergraduate students. How do introduce the order of an integer modulo n and primitive roots? How do I make this a motivating topic and are there any applications of this area? I am looking at something which will have an impact.

 

[Olinguito]

I would say build it up slowly, using concrete examples they can all understand. The simplest example would be clock arithmetic as time is surely something they must all deal with every day.

Though the students are undergraduates, let’s assume they are starting at the very basic level, with no prior knowledge of the subject whatsoever. Start by asking them the simplest of questions, say: “It is 9 o’clock now and I have an appointment in 5 hours’ time. What time is my appointment?” Most of them should have no difficulty working out the answer themselves. Now ask them a slightly more complicated one: “If it’s 9 o’clock now, what time will it be in 50 hours’ time?” Now they have to subtract a multiple of 12 rather than just 12. Explain to them that this is the principle of modulo-12 arithmetic.

Now tell them we can do modular arithmetic in numbers other than 12. For example, if they have to calculate not just the time but the time of the day (i.e. whether it’s a.m. or p.m.) then they’re working modulo 24 rather than 12. Another example would be modulo 7 for days of the week: “It’s Monday today; what day of the week will it be in 100 days’ time?” Give them a few similar examples to work out for themselves. They should begin to grasp the connection between the arithmetics of modulo 12, 24 and 7.

Once they have the right feel for the topic, it’s time for serious work. Define modular arithmetic formally, derive the basic laws (if $a\equiv b\pmod n$ and $c\equiv d\pmod n$ then $a+b\equiv c+d\pmod n$, and so on) and establish other important results. This should gradually build up to increasingly complex topics in modular-arithmetic number theory.