nontrivial in the elementary.

i think i have this phobia of being boring, especially as an instructor. it's why i try to bring up unconventional topics in the classes i teach:

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in my "proofs" class this semester, we were going over proof by contradiction. one classic theorem is why there are infinitely many primes;

so during the lecture,

1. i define the notion of a prιme number,
2. i give a few quick examples,
3. i state the theorem,
4. then i state the prιme number theorem to give them a sense of how the prιmes are distributed amongst the natural numbers,
5. and i tell them about the twιn prιme conjecture: despite the asymptotic distribution given in the previous theorem, nobody (yet) can explain why these consecutive occurrences seem to appear infinitely often.

i don't think there is any loss in showing students what is important (but easy to formulate) and what we don't know. statistically, they won't all be research mathematicians ..

.. but then again, isn't it worthwhile to show them why maths is interesting? isn't that the point of this sort of class?.. other than imparting the notion of rigor, i mean ..

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anyway, we start set theory today.
maybe i'll explain russell's paradοx to them. (-: