Prime numbers go on forever

Number theory is strange. Math is a hard enough subject because you can't just memorize facts and regurgitate them later. Trying to convince a young, dumb kid that math is useful is hard enough without a entire discipline devoted to studying numbers. Not how numbers show us the beauty of the universe, not the patterns the the planets make as they dance around the sun, or even how numbers can predict life events. No, it just studies. . . numbers. It studies their relationships to each other, and maybe there is infinite numbers? Or not? There might be a reason why one pattern in a set of numbers is kind of similar to another? Again, maybe not? Maybe it's just a coincidence.

Prime numbers though, those are fascinating. Prime numbers are the first introduction a student gets to the idea that something isn't right in the world of math. That math isn't perfect. It's an easy enough introduction to the world of number theory, since all it requires is the ability to factor numbers.

A quick refresher: Prime numbers are numbers that can only be produced by multiply 1 with the prime. 3 is prime since 1 * 3 = 3, but no other real integers multiply together to make 3. 6 is not prime because it has the factors of {1, 2, 3, 6} or 1 * 6 and 2 * 3. This is kindergarten stuff, really. The world of primes is much, much deeper than that though, and this month I'll be spending 3 days a week to talk about different subjects of primes.

The earliest work into primes comes curiosity of our old friend Euclid. He was the man who proved there are infinite primes. Working with the small primes we know, {2, 3, 5, 7} if these are multiplied together, the product is 210. Add 1, and the number becomes 211. Why add 1? Well, 210 can be factored by the primes in the list, but can 211? Nope. {105.5, 70.33, 42.2, 30.1429} is the list of numbers when 211 is divided by these primes. If a number can be factored, it will be factored by a prime number. If 211 cannot be factored by the primes in the list, then there is 2 conclusions: it can be factored by a prime NOT on the list of small primes we know, or 211 is prime number. Either way, the list is incomplete.

I'll start smaller. p = {2, 5}. 2 * 5 = 10; 10 + 1 = 11. 11 is a number I happen to know is prime, but the proof is easy, divide by every prime up 11, and you'll find they don't work. So the list becomes p = {2, 5, 11} 2 * 5 * 11 = 110, 110 + 1 = 111. Again, this cannot be divided by numbers from the list p so again the conclusion can be drawn that the list is incomplete.

The actual theorem states that given a finite list p of prime numbers, {p₁, p₂, p₃, . . . p_n} these numbers can be multiplied together with 1 added to the result. This number, p + 1, cannot be factored by any of the previous primes, so the original list is incomplete. This number is also prime, and by repeating this process ad infinitum the result is more prime numbers.

I'll be talking about primes three times a week for the next few weeks, until September 27 when the Philly Math Meetup will talk about primes. For now, I'm not so sure how to get that group up here for everyone not in Philadelphia to see, but I am working on a soundcloud thing for now. It just needs to be edited and uploaded. More primes on Wednesday, and an IWSG. Tomorrow, intro to bash script.

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EDIT:

This is more geared towards my animal goal update, but check out this page here for some dogs in Moscow that need help with food and shelter. Page is in English. Long story short, some dogs were found and are to be euthanized. Anyone can help by donating something to help with the care for these dogs.