The square is a pretty interesting place to start, though, since we already know there's a relationship between quadratic polynomials over the naturals and the distribution of primes. These polynomials describe lines that (eventually) become straight.
This is definitely amateur math hour for me. I was a math major a few years ago, but I did not study any number theory.
Maybe it's time to dive deeper into Objective C and / or OpenGL. I've got a few ideas that would be fun to try in real-time, but most of the more "fun" programming languages have poor / slow graphics libraries, and something low level lets you handle millions of entries without breaking a sweat.
Usually I'm struck by how unbelievably awesome the UI is, and how scores of similar attempts have been incomprehensibly awful. Meanwhile, Apple comes out with QC, a fast, elegant, simple version that goes almost totally unnoticed. Weird.
He set up the matrix of all possible integers, and started his computer stringing the primes across its surface as beads might be arranged at the intersections of a mesh. Jeserac had done this a hundred times before and it had never taught him anything. But he was fascinated by the way in which the numbers he was studying were scattered, apparently according to no laws, across the spectrum of the integers. He knew the laws of distribution that had already been discovered, but always hoped to discover more."
chop: cut with a hacking tool
cut away; "he hacked his way through the forest"
a mediocre and disdained writer
If you're at least a bit like me (no deep knowledge of maths, but growing curiosity) I recommend checking out BBC's recent documentary on primes, called 'The Music of the Primes'. It'll spark your interest, and it'll make you want to dig a little deeper.
BTW I hadn't seen that website until I went looking for a URL for the book just now. The site seems to have some pretty interesting material in its own right too.
If you don't know her work, you should. She's so much fun...
Can you imagine a world where someone discovers a way to trivially decode every https or ssh session on the internet? I fear we are building a city on top of the fog.
http://www.divisorplot.com/6.html -- the Ulam spiral
If this mathematics teacher had had access to the diagram of the Ulam spiral, however, I imagine that this alone could have provided the impetus for at least one student - me - to have made an academic career out of mathematics. As it is, she committed many other crimes against education, including her infamous catchphrase "Don't bother studying any kind of pointless mathematics that you'll never need to use at work." A catchphrase whose validity has been refuted many times over the years, not the least with the discovery of this diagram.
In other words, I'm struck by the ability of mathematics to generate such apparent complexity from simple principles :)
A pair of mechanical gears with relatively prime numbers of teeth have a longer service life: a given tooth rubs the same amount against every tooth in the partner gear. If they are not relatively prime, the hardest tooth would hit only a few teeth on the partner gear, wearing them out many times faster.
:) See also: http://en.wikipedia.org/wiki/Emergence
Come on, really? It's an intriguing visualization, but the patterns (in so far as they are real, and not an artefact of the limited size of the spiral) can be explained with some high school math.
EDIT: To hurry along the conversation a bit - it's entirely possible I've been misled into thinking things are more simple than they really are. Wouldn't be the first time...
The question is why certain lines have lots of primes and not others? why is a given polynomial so rich in primes, while other similar ones are not?
That's what some of these conjectures are trying to prove.
Huh? If you draw the Ulam spiral on a checkerboard, all odd numbers end up on squares of one color and all even numbers of squares of the other color, but that is neither necessary nor sufficient to get those diagonal streaks in the picture.