The mystery of the primes

Will we ever know how prime numbers are distributed along the number line?
28 September 2018


On Monday 24th September, Sir Michael Atiyah - winner of the Abel Prize and Fields Medal, former master of Trinity College, Cambridge and former President of the Royal Society - spoke on a potential proof of the Riemann Hypothesis at the Heidelberg Laureate Forum.

The Riemann hypothesis is one of the six remaining Clay Millennium Problems, seven problems set in 2000 with a $1 million dollar prize for solving, but incredibly was also previously on a list of 23 century defining problems laid out by prominent mathematician David Hilbert at the International Congress of Mathematics in 1900. Hilbert thought the problem so difficult and important, he is noted to have said: “If I were to awaken after having slept for a thousand years, my first question would be: has the Riemann Hypothesis been proven?”

A solution for the Riemann Hypothesis also happens to be what Russell Crowe’s character tries to present a solution of in “A Beautiful Mind”. The problem surrounds the fact that no one can predict what the next prime number will be, as they do not follow a known distribution pattern. The initial suggestion for a mathematical solution to this issue was proposed by Riemann in 1859, and has in fact been checked for a great deal of prime numbers. Though it has so far proved to be correct for those, the hypothesis has not yet been proved to work for all possible prime numbers.

Additionally, the largest prime number known already has 23,249,425 digits. That’s not a typo; it really has over 23 million digits. So why do we need to know more about primes when we already know so many, and the largest is almost inconceivably long?

Well prime numbers have an important part to play in both pure mathematics and applied fields, particularly in data encryption. To protect our data but still allow people to message us, a public key is given out freely, which is a very large number, and this is used to encrypt or “lock” the information. This very large number is actually the multiplication of two large prime numbers, which together are known as the private key. No one else has these two numbers, and they are then used to decrypt or “unlock” the message once received; attempting to use any other numbers would result in gibberish.

So is Atiyah’s proof correct? Well this will be the really tricky part. The proof he presents is based upon as yet unpublished work on something called the Todd function, an equation developed in theoretical physics. Without the chance to yet scrutinise the full works, mathematicians are unable to commit to comments on its validity.

In addition, there have been many attempted proofs over the years, and, under scrutiny from peers, all have either proved to be incorrect or true only within certain limits. So the best we can say for now is maybe it has been solved, maybe it hasn’t. But whatever the case, it has people talking about the mathematical problems of our age with seldom seen excitement, and isn’t that a worthy enough achievement on its own?


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