Why Primes?

"Upon looking at these numbers, one has the feeling of being in the presence of the inexplicable secrets of creation." D. Zagier from "The first 50 million prime numbers", The Mathematical Intelligencer 0 (1977) 7-19.

Currently, the largest known prime is 2^6972593-1 (A Mersenne Prime). Because it can take months to factorize numbers this big , it is a slow process finding new ones... But can we make good guesses at where they are?

Are the primes distributed chaotically or can we find some means for computing them? - Will it ever be possible to predict with arbitrary accuracy where the next one lies? I think either a yes or no answer yields a significant consequence.

If yes, RSA encryption is in big trouble for one. Large primes are vital to cryptography. RSA encryption is based on elementary mathematics involving prime numbers. There is no better mathematical way to protect data security. For a simple explanation see The Mathematical Guts Of RSA Encryption. Tantalizing is the fact that RSA solely relies on the present reality that there is no easy way to factorize these numbers, and that it hasn't been proven that there isn't another way.

If no, I wonder what purpose these numbers have. Primes are endemic to the counting numbers and the counting numbers are invariably fundamental to the development and maintenance of any intelligent civilization, that might spring up anywhere in the universe. Do the primes encode some message from the creator of the number?- Perhaps they are vital coefficients in string theory.. There are some very interesting articles (String theory, quantum cosmology etc) relating to primes listed at Surprising links between prime numbers and physics .

NeXT Chief Scientist Richard Crandall patented the Fast Elliptic Encryption system based on Mersenne primes. The Electronic Frontier Foundation has also offered $100,000 US to anyone who can find the first 100 million digit prime!

 

A Better Starting Point

For an excellent resource for just about anything prime related, check out Dr. Chris Caldwell's Prime Page at http://www.utm.edu/research/primes/ .

 

 

Why Visualize Primes?

A good way to start to see the true nature of the series is when we bring the primes together and work with them as a they converge to take some shape. Like looking at the dots of a low resolution newspaper picture individually, the secret nature of the primes is embedded in the their distribution, not in the elements themselves. This can be related to many things in life. Our own bodies might be described as little more than a group of inanimate particles, forming a very complex structure. Like a beautiful piece of music picked apart one note at a time, the ingredients of a cake before we start making it, meaning is in the pattern, not in the constituents...

We cannot bring all the elements here together as the prime series is infinite, but we can certainly represent a finite portion of them graphically, and observe the way in which it grows as we add more - and by doing this, we can watch for fracticalities within the series - signatures of some order and design.

Graphical experiments can often produce very unexpected results and lead to further investigation. e.g.. The Prime Spiral.