Sophie Germain

Sophie Germain was a French Mathematician from the 19th century. Sophie isn't a strange name for 19th century french boys, she was a female mathematician. Women in math during the 1800's were very rare, and they were very amazing, to say the least. Today, I want to talk a bit about her life. But to many naritives focus on how hard women had it in the 19th century, not enough focus on her work. So Wednesday will be focused on her work, which deals with prime numbers.

There is two women during this time that I've done research on, and the other is a Russian women named Sophia Kovalevskaya, who became the first first woman professor of mathematics in Europe. I bring her up, because to be a woman and a mathematician you couldn't just have ambition. You also needed to come from an upper middle class family, have plenty of family friends ranging from liberal to radical, and live in a country during a time of intense political and cultural change. Kovalevskaya was mid to late 19th century Russia during the period when nihilistic and anarchist writers were forming the ideas that later inspired the Russian revolution. Germain was born in 1776 and lived during the French revolution, the Reign of Terror, and the Napoleonic Wars. Tough times make great people, I guess. They still need access to the best ideas and education.

Speaking of access to the great ideas, Germain's initial education in mathematics was by getting texts and notes of lectures from students at the Ecole Polytechnique.  This worked in her favor, as she began to come up with her own ideas and submitted a paper to the math historian, Adrien-Marie Legendre, under a boys name. He was impressed with the work and was surprised to find that the work came from a woman¹.

Her real break came from the fact that she was able to keep in contact with the leading mathematicians of the day. Namely, she was able to initiate contact with Carl Friedrich Gauss (something like getting into contact with Stephen Hawking, then getting his help with problems). She used her correspondence with both men to work on her ideas of number theory, then later elasticity.

In her later years, she discovered she had breast cancer. She still worked despite the pain, then died at 55. She was not listed as a mathematician at her death, but merely "property holder". She never married, had know children, but after her death her friend Gauss was able to get an honary degree for her.

Wednesday, I'll focus on her work on prime numbers in number theory.

Back from a long weekend.

Did the Kennett Square Mushroom Festival all weekend. Good times, but spent. Hope your weekend was good.

On to week 2

Well, week one done. What makes this harder than the A to Z challenge and some other posts is that this is more than 1000 words. I really need an editor. In fact, that would be really nice to have for this damn thing. It would keep me from knowing how far off topic I got, it would keep my writing tighter, and I wouldn't ramble as much. Now if I could find a friend who could do math that would be willing to read my stuff.

I ended up missing two posts this week. The things I thought would get in the way, did. Since starting this blog, I've gotten much better at time management, but that still doesn't stop life from getting in the way. And sometimes I just can't things done if I have somebody walking up to me every 10 minutes asking me things. The best thing to try then, is to write some outlines of the posts I want to do this week today and tomorrow. That way, I can just add better wording and formatting and it should be quicker to write.

Yesterday I was going to do Belphegor's Prime, but as I was working on it and kept getting stopped it got way too late. I'll move that post to next week, then. Monday will be about Sophie Germain, a french mathematician I've wanted to write about for a while, Wednesday will be Sophie Germain primes, and more about Happy Primes, and then Belphegor on Friday. Tomorrow I'll figure out the other two days. Happy Saturday.

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.

IWSG - But what DO we do with the body?

Blogging is dead, therefore we should kick and desecrate its corpse. Dress it in fancy clothes and a silly hat, then hang it in the window of the downtown shops. Don't worry, that sentence doesn't cross any lines because nobody shops downtown anymore. They can't be offended if they can't see it.

Now that blogging is dead, maybe we can focus on writing? I mean, blogging left behind some pretty good stuff, and I'm sure he wouldn't mind if we took it (Blogging is masculine, btw, because it ends with a "g" not an "a" which is feminine or an "e" which is gender neutral.) The blog hops, blog fests, writers groups, and general community vibe is a pretty nifty thing. This whole group of writers have banded together to help out each other get published and advertise. People can actually make money by posting journal entries on the web for others to see! It's suggested I should post about about a topic, but come on. I've come across plenty of blogs with no original ideas, no concept of language, and the whole thing looks like it was designed with mspaint on a broken tablet. In fact, you're reading such a blog right now! These guys still have books published though. Maybe it's a good thing blogging finally died. It was starting to get grotesque and boated, and its gangrenous parts were starting to stink.

This internet thing is pretty snazzy medium. Like nothing that came before it, and perhaps like nothing after it. I don't have to learn another language, because my browser tranlates stuff automatically for me. Any of you out there read mspaint adventures? What always impressed me about it, before it became 3000 pages long, is it used the medium to its advantage. Everything is possible; sound, writing, pictures, video, games. I know one of the things lacking in my writing, besides focus and good spelling, is that it doesn't use the formatting ability. Italics, bold, and underlining can emphasize things, but sizes of words can denote volume as well.So writing can become a symphony, with sentences read at a pianissimo, then they can explode into a thunderous forte! Sadly, I still can't realize that dream on blogger. Well, work within the limits,

Insecurities today? I'm reading your comments, and I'm wondering  how I can talk math without using numbers and calculations. It's very possible, you know. The greeks were very picture and geometry oriented,then i the middle ages mathematical answers were passed along using poems. But now, if you can't decpher the symbols and calculations, you're cut completely out of a very elegant language. It's something to play with while I talk primes. My prime post is going up, later! This is quicker and simpler, sort of, and it's going up first. Now for some goals. Next month I want to say "Hey guys, I picked out my 7 to 10 favorite blog posts, and over the next year I'll work on editing them so they are able to be published!" That's my goal. Happy blog hopping.

Hello, World

Ok, ok, ok. I've got a busy couple of months, so I'm going to pass the stuff off onto this blog. Monday, Wednesday, and Fridays, prime number posts will go up. Tuesdays and Thursdays, bash script. This blog is one of the best ways for me to learn and become better at a topic. I'm actually really bad with school, surprisingly enough.

Maybe you've heard about some sort of "Linux" thing or maybe some weird "Ubuntu" thing that the dorks and nerds are talking about these days. Maybe you've heard it's free, and uses free software, or maybe you decided that the mascot, Tux the penguin,

Tux the Penguin, cuter than an Apple

  is cute, as opposed to Windows bright color thing or Apples' apple. Maybe that made you download and then you installed it with help of "the computer person" in your family. And now it's on the computer and. . . what the fuck is the big deal?

Really, this Ubuntu thing is brown. It doesn't seem to do anything different except make some task that were once easy on a Windows harder. All you keep hearing about is "How it's so easy to customize!" Well, you can change the background and the theme, whoop-ti-frickin'-do. Could do that on a PC or a Mac, and those worked 90% of the time!

Welcome to the wonderful of bash scripting. It's not much in the world of customization, not much at all really. But it's an easy place to start, and once a person get a hang of it, well I reckon that person would have all the power in their hands. And since it's a Unix thing, Mac users can join in on the fun! Windows users have to deal with the fact that Microsoft keeps pushing their own technology and it doesn't play nice with each other. Windows users have to deal with the fact that they are like Mac users in the 90's.

Lets start. Open the program called "Terminal Emulator", which should be under accessories. You'll be looking at a box that has this in it:

[username@comp ~]$

Type in the following command:

echo "Hello World"

Yup, it's the good 'ole hello world program. All it does is display the words "Hello World". For something functional, you could backup your directory file. Simple, easy, and all you need to do is type:

mkdir Backups

That makes a new Backup file. Then type:

tar -cvzf Directory-mybackup.tar.gz Backups

Now if you accidentally delete the whole directory file, you have a backup! That's it for day one. Come back Thursday for  more. And if I have time this weekend, come back then for a rant on why people should learn programming, and not just Visual-Basic. Oh, and speaking of this weekend, there is this thing called the Kennett Square Mushroom festival in Chester County, Pa. I'll be there this weekend, answering questions about how to grow mushrooms. Stop by, say the secret word (the secret word is: swordfish) and the duck comes down!

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.