Thursday, September 4, 2014

Types of Primes

Types of primes! Thought primes were just simple elements of multiplication? Nope, there are different types of primes, and it's what changes primes from simple number theory to number game.

For the writers and the word nerds out there, there are palindromic primes, or palprimes. It's exactly what it says on the box, they are primes that are the same backwards and forwards. 11 is palprime. 919 is palprime. My obsession, 1000000000000066600000000000001, is palprime, but I'll talk about that more tomorrow. There are plenty of programming challenges out there for writing script that produces prime numbers. It's a good challenge, really. Given an integer, produce the smallest prime that is also a palprime greater than the given integer. It's a good challenge, because not only would you have to check and see if the number has any factors, but the script would also have to check and see if it's the same number forwards and backwards. If I can get back on my schedule today, I'll take a crack at it with bash. Don't count on it though.

Numbers can be happy, and so can primes. Start with an integer, any integer. (I keep saying integer. Just pick a number that's not negative, is not a fraction, and doesn't have decimal) Say, 44. First, square both numbers, so 42 and 42. Add them together, 16 + 16 to get 32, then repeat. So, 32 + 22 = 13, and 12 + 32 is 10 and 12 + 02 is 1. If the sequence ends in one, then it is happy. If it doesn't end in one, then it loops forever. A happy prime is a happy number that is prime. The first 5 are 7, 13, 19, 23, 31. Feel free to work out the rest.

 Visual Representation of Eisenstein Primes
There are Eisenstein Primes. I'm putting this at the end, otherwise everyone's eyes would glaze over and get out of here too early. Eisenstein Integers are integers with the form of Z = a + bω where ω is equal to: ${e}^{2\mathrm{\pi i}/3}$ Heh. Look at that. π and an imaginary number! It is important in higher math, but hey, look at that  Z = a + bω again. That kind of looks like a complex number (not an oxymoron), a + bi. Complex numbers are just a larger equation for (x, y), which is coordinates on a number plane. Which means you can plot Eisenstein primes. Makes a pretty picture, doesn't it?

Primes act in strange ways. Honestly, there are many more types of primes, because primes are the elements of multiplication.

Stupid Wednesday threw me off my schedule, but whatever. The point of this month is partly to prepare for next month, so I'll work around busy stupid Wednesdays. Argh. Tomorrow will be Belphgor's prime, so If I can't do bash today, then bash will work into what I want to do tomorrow.

1. You're...making...my...brain...grow... My skull cannot handle the new pressure; it can't expand!

I was okay till you got to the Eisenstein primes. Here's how a Cherdo brain works: if I can't see exactly what is happening in my head, my cranium freezes up. Now, I can memorize slews of info. That's only a short term thing. For true knowledge that sticks...gotta see that picture in my head. Darn if you aren't the only person thus far to provide a picture for me!

I'm not smart enough to follow your blog (but I will anyway).

1. I'm not that smart, to tell the truth. I just hang out around people much smarter than I am who are interested in the same things. Eisenstein Primes are nasty, because they combine the craziness of imaginary numbers and primes. To do a topic like this justice, it would need a way longer post. For now, know that imaginary numbers (i) work in the equation x + yi = (x, y). That's a coordinate on a plane. 3 + 5i would just be the point (3, 5). Eisenstein decided to examine the cube roots of 1 and their complex roots. So instead of using i which is the square of -1, the Eisenstein numbers use cube root of 1. http://www.regentsprep.org/Regents/math/algtrig/ATO6/lessonadd.htm is how you add complex numbers, so they have rules for math as well. Eisenstein primes are the same as primes in that other Eisenstein numbers can factor them. I tried to find a good page for this, but http://www.decodedscience.com/the-complex-tale-of-eisenstein-prime-numbers/4751 is all I can find, and the way he does equations is really confusing.