If you’re anything like me, you haven’t heard of the Millennium Prize Problems – like so many other students, I stopped fraternising with anything remotely numerical after GCSEs. That doesn’t mean, however, that people like me are right in our ignorance of the existence of the Problems. As meaningless as they seem to those not revolving in mathematic spheres, the seven problems are of huge significance to the world as we know it.
The Millenium Prize Problems are unsolved problems that underpin vast fields of study within mathematics.
The Millenium Prize Problems are unsolved problems that underpin vast fields of study within mathematics. If solved and proved, they threaten to change the landscape of mathematics, and the way it operates in our lives – forever. The Clay Mathematics Institute established them, offering a $1 million prize to anyone who could solve and prove them. To date, just one of them has been solved (the Poincaré Conjecture solved by Grigori Perelman in 2003).
Now, one mathematician claims to have proved a second of the Millennium Prize Problems. The Riemann Hypothesis, first put forward in 1859 by German mathematician Bernhard Riemann and left unproved since then, is a prime number theorem. For those of us without an A-level in maths, the explanation is this: the theorem revolves around how prime numbers don’t follow a regular pattern – they occur completely randomly. The Riemann hypothesis suggests there’s a way of finding them. Riemann identified that how often prime numbers occur follows, very closely, one equation, known as the Riemann Zeta function. This has been tested and proved correct on the first 10,000,000,000,000 (ten trillion) prime numbers but is yet to be proved to infinity.
For those with a higher understanding of mathematics, the explanation on the Clay Mathematics Institute website states this:
“The prime number theorem determines the average distribution of the primes. The Riemann hypothesis tells us about the deviation from the average. […] all the ‘non-obvious’ zeros of the zeta function are complex numbers with real part ½. […] the frequency of prime numbers is very closely related to the behavior of an elaborate function:
“ζ(s) = 1 + 1/2s + 1/3s + 1/4s + … The Riemann hypothesis asserts that all [non-trivial] solutions of the equation
“ζ(s) = 0 lie on a certain vertical straight line.”
Michael Atiyah, an 89-year-old mathematician emeritus at the University of Edinburgh, makes claim to having proved Riemann’s hypothesis. He has done this through a method called “contradiction”, where he proves a scenario is impossible if the hypothesis is not true, by describing “the fine structure constant”, a quantity in physics that describes electromagnetic interaction between charged particles, with the Todd function.
It’s all very well and good having so much technical jargon in one place, but what are the real, practical implications of it all? Turns out, encryption (and consequently protection) of any and all online data relies solely on the unpredictability of prime numbers. Everything you put online – from the name and address you submit for your online order to your card details that pay for it – is protected from those whom you don’t want to see it by the encryption that prime numbers offer in the form of the RSA algorithm. If the hypothesis proves true, all the encryption keys that have been in place to secure online information will become readily available to anyone who wants them.
Maybe it’s fortunate, then, that many critics are sceptical of Atiyah’s proof. They cite his previous incorrect claims of proof in earlier years. Moreover, Atiyah’s proof relies on the Todd function having certain properties that, critics claim, couldn’t possibly be attributed to one function; there is no existing function with the properties he claims. Atiyah is an authoritative figure in the field of mathematics, however, having already been the recipient of two prestigious prizes in mathematics, the Fields Medal and the Abel prize, his proof is being taken seriously despite its sceptics.
Only time will tell if Atiyah has really answered the million dollar question. Regardless of whether his proof is correct and whether our online information stays safely encrypted or not, one thing remains certain: our understanding of mathematics will never stop evolving, and the implications of it on our lives will only become more and more significant.
By Georgie Wardall
image source: http://whatwhenwhy.net