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## Thursday, May 21, 2015

### An Intentional and an Unintentional teaching experiment regarding proving the number of primes is infinite.

I taught Discrete Math Honors this semester. Two of the days were cancelled entirely because of snow (the entire school was closed) and four more I couldn't make because of health issues (I'm fine now). People DID sub for me those two and DID do what I would have done. I covered some crypto which I had not done in the past.

Because of all of this I ended up not covering the proof that the primes were infinite until the last week.

INTENTIONAL EXPERIMENT: Rather than phrase it as a proof by contradiction I phrased it, as I think Euclid did, as

Given primes p1,p2,...,pn you can find a prime NOT on the list. (From this it easily follows that the primes are infinite.)

Proof: the usual one, look at p1xp2x...xpn + 1 and either its prime or it has a prime factor not on the list.

The nice thing about doing it this way is that there are EASY examples where p1xp2x...xpn+1 is NOT prime

(e.g., the list is {2,5,11} yields 2x5x11 + 1 = 111 = 3 x 37, so 3 and 37 are both not in {2,5,11})

where as if you always use the the product of the first n primes then add 1, you don't get to a non-prime until 2x3x5x7x11x13 + 1 = 30031 = 59x 509.

They understood the proof better than prior classes had, even prior honors classes.

UNINTENTIONAL: Since I did the proof at the end of the semester they ALREADY had some proof maturity, more so than had I done it (as I usually do) about 1/3 of the way through the course.

They understood the proof better than prior classes had, even prior honors classes. Hence I should proof all of the theorems the last week! :-)

But seriously, they did understand it better, but I don't know which of the two factors, or what combination caused it. Oh well.

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Starting with {2}, and repeating the process do you eventually get all primes? Is *any* finite set of primes, such that the process generates all primes?

ReplyDeleteI started my algorithms class this Summer with a very similar technique to the one you described, with the intention of showing them that coming up with algorithms has more purposes than making computers do things fast. I also think it went quite well!

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