The British-Lebanese mathematician Sir Michael Atiyah spoke at the Heidelberg Laureate Forum on 24th September. In a 45 minute talk he claimed to have found a “simple proof” to the Riemann hypothesis, a problem that has remained unsolved since 1859. Correct proof to support the hypothesis, labelled by the Clay Mathematics Institute as one of the seven “Millenium Prize Problems”, could have huge implications for the majority of modern day cryptography, including cryptocurrencies like Bitcoin.
The hypothesis concerns prime numbers and the ability to find the number of primes smaller than any given integer, N. The hypothesis relies on the Riemann zeta function and its return of zero. It is known that zero is returned when a negative integer is used in the function, these are known as trivial zeros. This is also the case with any complex number whose real part is ½, known as non-trivial zeros. The non-trivial zeros have varying imaginary units but consistent ½ values, allowing for their calculation.
However, the continuing problem is the lack of proof that complex numbers with a ½ real value are the only form of non-trivial zeros. Until now this has been assumed to be true and has provided the basis for modern cryptography. This is due to the property of prime numbers where calculating the product of two primes is simple but finding the two primes used when the only information given is the result is very difficult. This allows for one-way functions that cannot be easily inversed by those that are not the intended recipient.
But proof of the hypothesis may lead to a connection being observed between prime numbers and this could be exploited to counteract contemporary forms of coding. This could lead to hash algorithms being easily hacked and bitcoin being mined at an exponentially faster rate.
Atiyah’s claim of proof is currently met with scepticism but if correct his work could have an enormous impact. Computer Science and Mathematics applicants can develop their understanding of the Riemann hypothesis and its underpinning of cryptography and its other applications.