Prime numbers and infinity

Yesterday I wrote about a really neat site showing patterns arising out of factoring integers:

A friend of ours on twitter, Federico Chialvo, made me aware of this site Friday on twitter.  As we watched the patterns play out yesterday, my younger son asked a nice little question –  how many prime numbers are there?  Fun question, of course, and I thought I’d use that as the topic for one of our  Family Math videos in the future.

Next up yesterday was a neat post from Patrick Honner about infinity.  Patrick is a high school math teacher in NY who seems to have an endless supply of interesting ideas about both math and teaching math:

His post reminded me of a curious comment in my college analysis book, “Principles of Mathematical Analysis” by Walter Rudin, or  sometimes “Baby” Rudin.  Problem 10 in Chapter 8 asks this gem of a question:

Prove that the sum of (1/p) diverges, where the sum extends over all primes.  (note – sorry I don’t know how to format math equations in the blog, yet).   Following the problem is an interesting comment from Rudin – “This shows that the primes form a fairly substantial subset of the positive integers.”    The comment struck me when I was in college, and the discussion in Patrick Honner’s post reminded me of it at a time when I already had the subject of why there are infinitely many primes in the back of my mind.

In any case, with that background, here’s what the boys and I talked through this morning:

Definitely a fun start to the day.