Win a million dollars with maths, No. 1: The Riemann Hypothesis

 

 

Huge primes are used to encrypt information. The Riemann Hypothesis calculates how many there are beneath a given threshold.

The first million-dollar maths puzzle is called the Riemann Hypothesis. First proposed by Bernhard Riemann in 1859 it offers valuable insights into prime numbers but it is based on an unexplored mathematical landscape. If you can show that its mathematical path will always lie true, $1m (£600,000) is all yours.
Mathematicians are obsessed with primes because they are the foundation of all other numbers. Prime numbers in mathematics are like atoms in chemistry, bricks in the construction industry and ludicrous pay cheques in professional football. Everything is built up from these fundamental units and you can investigate the integrity of something by taking a close look at the units from which it is made. To investigate how a number behaves you look at its prime factors, for example 63 is 3 x 3 x 7. Primes do not have factors: they are as simple as numbers get.
They are simple in this one respect – but are otherwise extremely enigmatic and slip away just when you think you have a grip on them. Part of the problem is that, by definition, they have no factors, which is normally the first foothold in investigating a number problem. This is also the key to their usefulness. It is their difficulty to grasp that makes primes the basis for our modern information security. Whenever you use a cash machine or visit a secure website, it is huge prime numbers that encrypt your information and make it extremely difficult for anyone else to pry it back out of the electronic cipher.
Prime numbers also have the annoying habit of not following any pattern. 3,137 is a prime and the next one after that is not until 3,163, but then 3,167 and 3,169 suddenly appear in quick succession, followed by another gap until 3,187. If you find one prime number, there is no way to tell where the next one is without checking all the numbers as you go. One possible way to get a handle on how primes are spaced is to calculate, for any number, how many primes there are smaller than it. This is exactly what Riemann did in 1859: he found a formula that would calculate how many primes there are below any given threshold.
Riemann's formula is based on what are called the "Zeta Function zeroes". The Zeta Function is a function that starts with any two coordinates and performs a set calculation on them to return a value. If you imagine the two initial coordinates to be values for latitude and longitude, for example, then the Zeta Function returns the altitude for every point, forming a kind of mathematical landscape full of hills and valleys. Riemann was exploring this landscape when he noticed that all of the locations that have zero altitude (points at "sea level" in our example) lie along an straight line with a "longitude" of 0.5 – which was completely unexpected. It's as if all the places in England that are at sea level (ignoring the coast) are on a dead straight line that runs directly north along the 0.5 longitude line.
Riemann used these zeroes as part of his prime distribution formula, but the problem is that no one knows for sure that all of the zeroes are on that same straight line. Sure, mathematicians have checked that the first ten trillion zeroes all fall on that line, but that's no guarantee that the ten trillionth and one zero might be somewhere else, throwing the whole prime distribution formula out the proverbial window, along with vast amounts of related number theory. Which is why there is a $1m prize for anyone who can show that all of the Zeta Function zeroes line up on the "0.5 line" without resorting to the impossible task of walking along this infinite line to check.
I've given you the Zeta Function to get you started and if you dust off a bit of "complex variable" maths, you will be well on your way to exploring the Riemann landscape. However – if that's a bit much – here is an easier starting problem: All prime numbers (greater than five) squared are one more than a multiple of 24. Check it for a few – it works. You can even prove that it works for all of the infinite number of primes.
Now if you can just do that for the Zeta zeroes, you can stop kicking a football around in the cold in hope of a big pay day.