Episode 4 - 13 January 2009

Humans find it perfectly natural to wonder at what makes up the stuff around us. When ancient mathematicians tried this, prime numbers were born. Primes are both stunningly simple to explain and stupefyingly difficult to understand. Any natural number (i.e. the positive integers: 1,2,3,...) can be written as the multiplication of prime numbers, or equivalently, every natural number has a unique prime decomposition. For instance, 30 = 2 x 3 x 5. A prime is a natural number that can only be divided by itself and 1, e.g. 7. A small caveat: 1 is not a prime number under modern definitions. Why, you ask? It makes other things much simpler if we forget about 1. So that's what prime numbers are, but understanding them frustrates the sharpest of minds. They seem to follow no obvious pattern: 19 is prime, but 21 isn't. There is no formula to tell us which numbers are prime, and which aren't.

We know that there are inifintely many primes, and we know roughly how many primes there are less than a particular number (I say roughly, because the estimate we have is not very good for low numbers, but gets better for much large numbers). But there are many problems which are unsolved. How many twin primes are there (primes separated by 2, e.g. 11 and 13)? How many Mersenne primes are there (Mersenne primes are 2P-1 where P is itself a prime. These are the biggest known primes.)? Can every even number greater than 2 be written as the sum of two primes (this is known as Goldbach's Conjecture)? And the greatest unsolved problem in mathematics, the Riemann Hypothesis (first stated in 1859), is intimately related to where the primes are. If we solve the Hypothesis, then we will know a lot more about prime numbers.

People other than mathematicians care as well. The RSA algorithm, which keeps everybody's money safe, relies on the difficulty of breaking numbers into their prime components, and cicadas use primes to avoid predators and disease. But these are just novelties compared to the implications of the Riemann Hypothesis: if true then the Hypothesis says that primes are inherently part of the way atoms and molecules all over the universe are hanging together.

You might say these numbers are of prime importance.

You can find information about primes all over the place (even the internet), maybe start with an introductory number theory textbook, or mathworld or even wikipedia. Here are some specific references:

BETA: Quantum mechanics says some truly weird stuff about the universe. Einstein himself, one of the founders of the quantum theory, refused to believe in many of its implications. Human minds just don't seem to be able to comprehend what really goes on at really small scales. Mathematically, it all makes sense, but if you try and put it into words to get a better idea of what is 'really' going on, then even the best scientists can get hopelessly entangled. And that appears to be what happened to the people who created the film What the #\$*! do we (K)now? This documentary attempts to explain psychic phenomena as a consequence of quantum mechanics. However, they make several unjustifiable leaps of logic. They say that everything in the universe is really just information, and in some sense that is true; everything is expressible as a density of the relevant field. But then they claim that this means stuff is the same as thought, and so we create stuff by thinking about it. At another point, they talk about 'collapse of the wave function', which, under the standard interpretation of quantum physics, is the process whereby a system goes from being a statement of probabilities about what state it will be in, to actually being in one of those states. According to our Bleep friends this therefore means that we have the choice of which state all our particles will be in. Both invalid arguments, quantum mechanics says nothing about consciousness in these circumstances.

The rest of the film goes on in this vein, often starting with accurate statements, and then leaping off into paranormal conjecture, while still claiming scientific rigour. Oh well.

The quiz. End.

• Some info about Question 3: Land, Michael. F. & Nilsson, Dan-Eric (2002), Animal Eyes, Oxford University Press, USA.

Oops. Mistakes we shouldn't have made but did:

A 300 digit number multiplied by another 300 digit number is roughly a 600 digit number, not 60 or even 90 thousand digits.

The Riemann hypothesis was first put down on paper in 1859, by Bernhard Riemann. It was listed as one of the 7 Millenium Prize Problems.

Nini mentioned that the highest prime factor of any number will be lower than the square root of the number. What she meant was that any pair of factors of a number will contain one smaller and one larger number than the square root, so when searching for prime factors of a number you only need to check up to the square root.

In Question 1 of the quiz, which asked about the amount of energy needed for the DeLorean to jump around in time, we were really asking about 'power' instead of 'energy'. The first two options were power, the second two were energy. We didn't make the distinction clear. Note that power is the rate at which energy is being used.