I’ve always had a love-hate relationship with mathematics. On the one hand I loved the concept, the premise of mathematics - to describe everything using a set of rules - on the other hand I hated the tedium of it… I loved to learn the concepts and theories, but I hated to sit down and do the sums, especially when there seemed no apparent point to it. That’s one of the reasons why I like programming - it allows one to solve to ‘real’ problems. Because of this skewed view of math, I love to read about mathematicians and the applications of mathematics even though I suck at it :-)
When I saw this article on Slash-dot today I was immediately intrigued. It deals with one of the more infuriating concepts of math for me - Prime Numbers. A prime number is a number that can be divided only by itself and one. The infuriating part is there is no apparent pattern to prime numbers - you had to memorize them!
According to the article there is a correlation between prime numbers and the energy levels in the nucleus of a large atom. Excerpts from the article:
“Riemann discovered a geometric landscape, the contours of which held the secret to the way primes are distributed through the universe of numbers. He realized that he could use something called the zeta function to build a landscape where the peaks and troughs in a three-dimensional graph correspond to the outputs of the function. The zeta function provided a bridge between the primes and the world of geometry. As Riemann explored the significance of this new landscape, he realized that the places where the zeta function outputs zero (which correspond to the troughs, or places where the landscape dips to sea-level) hold crucial information about the nature of the primes. Mathematicians call these significant places the zeros. But then Riemann noticed that it did something even more incredible. As he marked the locations of the first 10 zeros, a rather amazing pattern began to emerge. The zeros weren’t scattered all over; they seemed to be running in a straight line through the landscape. Riemann couldn’t believe this was just a coincidence. He proposed that all the zeros, infinitely many of them, would be sitting on this critical line—a conjecture that has become known as the Riemann Hypothesis.
But what did this amazing pattern mean for the primes? If Riemann’s discovery was right, it would imply that nature had distributed the primes as fairly as possible. It would mean that the primes behave rather like the random molecules of gas in a room: Although you might not know quite where each molecule is, you can be sure that there won’t be a vacuum at one corner and a concentration of molecules at the other.
For mathematicians, Riemann’s prediction about the distribution of primes has been very powerful. But despite nearly 150 years of effort, no one has been able to confirm that all the zeros really do line up as he predicted.
It seemed the patterns Montgomery was predicting for the way zeros were distributed on Riemann’s critical line were the same as those predicted by quantum physicists for energy levels in the nucleus of heavy atoms. The implications of a connection were immense: If one could understand the mathematics describing the structure of the atomic nucleus in quantum physics, maybe the same math could solve the Riemann Hypothesis.
Mathematicians were skeptical. Though mathematics has often served physicists—Einstein, for instance—they wondered whether physics could really answer hard-core problems in number theory. So in 1996, Peter Sarnak at Princeton threw down the gauntlet and challenged physicists to tell the mathematicians something they didn’t know about primes. Recently, Jon Keating and Nina Snaith, of Bristol, duly obliged.
There is an important sequence of numbers called “the moments of the Riemann zeta function.” Although we know abstractly how to define it, mathematicians have had great difficulty explicitly calculating the numbers in the sequence. We have known since the 1920s that the first two numbers are 1 and 2, but it wasn’t until a few years ago that mathematicians conjectured that the third number in the sequence may be 42—a figure greatly significant to those well-versed in The Hitchhiker’s Guide to the Galaxy. It would also prove to be significant in confirming the connection between primes and quantum physics. Using the connection, Keating and Snaith not only explained why the answer to life, the universe and the third moment of the Riemann zeta function should be 42, but also provided a formula to predict all the numbers in the sequence. Prior to this breakthrough, the evidence for a connection between quantum physics and the primes was based solely on interesting statistical comparisons.”