Another exploratory moment

Okay, so I'm not giving an update (yet) on how Newton v. Leibniz went down...I've just remembered that I'd meant to spend a few moments this weekend chronicling last Friday's 280 meeting, in which we took a detour from my planned path and did a little mathematical bushwhacking.

Given numbers n and k, let Z[_k] be the set of integers modulo k, and define f_n from Z[_k] to Z[_k] by f_n(x) = xⁿ mod k.

The questions I'd planned to ask: is does f₂ = f₃ hold when k = 2? This question was quickly dispatched, as was the corresponding question when k = 3.

From here, it was impromptu math.

Quincy and Uri and a handful of others were curious enough to press the matter further: suppose k = 2 and we want to know when f_m = f_n; what then?

This question too was a simple one, and was easily answered.

What about when k = 3?

Not a problem: noting that 3 = -1 mod 3, we were able to tackle this question as well...

...the course had become an expedition, we were making up mathematics as we went along.

I felt a bit bad for several people in the class, since it seemed as though four or five individuals in the class were deeply involved in the discussion while the others were relatively disinterested onlookers.

There were no objections, though.

It was exciting!

Now I've got to go again...I've now got most of the final drafts of the N. v. L. arguments, and I'll be taking those home tonight to look over them.

It's been a lot of work on all of our parts. I hope this project has proven meaningful to the students. I look forward to reading their reflections, due next Monday.

Ciao!