'Greatest Unsolved Problem in Mathematics' May Have a Solution

The Riemann Hypothesis is one of the most intractable problems in mathematics — thus the lure of money. 3Blue1Brown/YouTube

A trio of mathematicians has set the academic world abuzz with a paper that offers a new approach to what has been called "the greatest unsolved problem in mathematics," proving the so-called Riemann hypothesis.

The paper, published in the journal Physical Review Letters, teases readers with this beguiling statement: "If the analysis presented here can be made rigorous to show that Η is manifestly self-adjoint, then this implies that the Riemann hypothesis holds true."

For the mathematically uninitiated, this statement probably seems hopelessly obscure. But for those in the know, it means big dollar signs — $1 million, to be exact. That's because proving the Riemann hypothesis is one of seven Millennium Prize Problems — widely considered the most intractable problems in mathematics — each of which carry a $1 million award for anyone who can solve them.

The Riemann hypothesis is named after the man who first proposed it, German mathematician Bernhard Riemann. The reason it's important is because it potentially offers a way to understand the distribution of prime numbers, which are notoriously difficult to wrangle; by all appearances, they seem to have a random distribution. If true, however, the Riemann hypothesis offers us a simple method to calculate how many primes there are below any given threshold, which would make the job of prime hunters extraordinarily easier.

The answer lies in ... quantum mechanics?

Perhaps the most tantalizing key to the new paper is that it proposes using quantum mechanics to solve the Riemann hypothesis. Yes, quantum mechanics ... that mind-bending branch of physics that purports to understand what's happening at the smallest scales of nature.

Basically, the paper's authors have proposed the existence of a quantum system where energy states correspond to the conjectured set of conditions in the Riemann hypothesis. Even better, they have defined a component, called a Hamiltonian operator (denoted as H), as being the crucial cog in such a system.

Assuming that everything checks out, this new method essentially reduces the gargantuan problem of the Riemann hypothesis to the much simpler problem of the Hamiltonian operator. That's what makes it all so exciting, and why proving the Riemann hypothesis suddenly seems within reach. Of course, the paper has to pass scrutiny first, which could take some time.

Still, it's an exciting time to be a mathematician, a prime number enthusiast, or even just an admirer of the human intellect.

You can learn more about the Riemann hypothesis with this incredibly informative video below: