In the modern world, pure mathematicians unfortunately do not make that much money because they are perceived as ‘useless’, working purely in the strive for a beautiful proof or an elegant derivation. I feel as if many areas of mathematics have gradually pushed their way into the realm of art, since they exist purely for aesthetics or intellectual curiosity. Of course, mathematicians do maths not for the money, but for the beauty. However, if you’re looking for a cheeky way to make a bit of quick cash in mathematics, there is one way…
At the start of the new millennium, the Clay Mathematics Institute in Peterborough set out a series of seven mathematical problems which had challenged mathematicians for generations. They were considered to be the most ‘important classic questions that have resisted solution over the years’. All of the questions are very theoretical and abstract in nature, which contributes to their difficulty because it requires an extremely diligent pure mathematician who can ‘think outside of the box’ to be able to solve them. For each problem, the first person to provide a solution to it would be awarded $1,000,000 as a reward. Now if that’s not an attractive incentive, then I don’t know what is. But I kinda lied earlier about it being easy. It is most definitely not easy. It is very, very hard.
So far, only one of the seven problems have been conquered. This was the Poincaré Conjecture, which states that “Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere”. In more simpler terms, “Any finite 3-dimensional space, which doesn’t have any ‘holes’ in it, can be continuously deformed into a 3-sphere”. This conjecture, or hypothesis, was proven by Grigori Perelman in 2003, and after review, was confirmed in 2006, which lead to him being offered a Fields Medal, the pinnacle of mathematical achievement equivalent to a Nobel Prize. Later, in 2010, Perelman was awarded the Millennium Prize for solving the problem and was offered the million dollar prize. On both occasions, he declined the offer. He believed that he did not deserve to be given an award for solving a problem that countless others had contributed towards. The Poincaré Conjecture had been worked on by generations before him, but he was just lucky enough to have been the one to finish. “I’m not interested in money or fame; I don’t want to be on display like an animal in a zoo.”
Out of the seven Millennium Problems, the Riemann Hypothesis is arguably the most important for number theory. It seeks to understand the most fundamental objects in mathematics – prime numbers. Prime numbers are the very atoms of arithmetic. Any positive integer can be made by multiplying together two or more prime numbers. Despite how fundamental prime numbers are for the basis of number theory and mathematics, we’ve never managed to find harmony in their weird, disjointed music. Every new prime number we discover seems to appear randomly, as if Nature chose it by flipping a coin. This random and unpredictable nature of prime numbers makes it the prime pursuit (see what I did there?) for mathematicians who have a desire to find order in numbers.
What the Riemann Hypothesis claims to offer is an incredibly accurate approximation for the number of primes under a certain integer, which would allow us to map the distribution of prime number a lot more efficiently. The proof of this Hypothesis is so important because so many other theorems rely on it. For the past 150 years, countless theorems have needed to say “if the Riemann hypothesis is true…”, so being able to prove it would immediately validate the consequences in these theorems as true. To go into any depth on the Riemann Hypothesis would prolong this post far too much, so I will leave it until another day.
You may be thinking, “What’s so special about these primes anyway?”
The fact that any number can be factorised into a bunch of prime numbers makes primes vitally important to modern communications. Most cryptography used in modern computers works by using the prime factors of large numbers. The large number used to encrypt the data can be publicly known, but in order to decrypt it again, only the prime factors of that large number can be used. Since prime numbers stay ever elusive, we do not have an efficient way to find the prime factors of very large numbers. For a hacker to compute the factors manually, it would take so much time that we say that it is impossible. A modern super-computer could chew on a 256-bit factorisation problem for longer than the current age of the universe, and still not get the answer. It is possible that as we develop new mathematical strategies or advanced hardware technology like quantum computers, we are able to prime factorise large numbers much faster, which would effectively undermine and destroy modern encryption.