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, {

*p*

_{1},

*p*

_{2},

*p*

_{3}, . . .

*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.

Not your fault, but I read posts like this one and feel kinda stupid. The fact that I was able to follow this one to the end (possibly) should have eased that feeling, but -alas- no. I have a feeling Prime Numbers are about to get more complicated and I will be officially Out To Sea.

ReplyDeleteI'm directing my students to this post. You made it interesting!

ReplyDeleteWhile I can write a somewhat coherent sentence or two, I'm number dyslexic. That makes me hyper-sensitive to the any dyslexic issue. ergo, my interest in @WeWrite4U_Lit on Twitter. I'm so pleased a number guy jumped in and signed up. Thank you very much. New follower!

ReplyDeleteBanned complain !! Complaining only causes life and mind become more severe. Enjoy the rhythm of the problems faced. No matter ga life, not a problem not learn, so enjoy it :)

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