What's the point of maths research? It's the abstract nonsense behind tomorrow's breakthroughs

Prime purpose

Research mathematicians essentially still discover similar types of results today. We have simply moved on to different questions that have become important, to new methods for existing questions, to different systems that draw our attention, and to more advanced questions about things that have already been researched.

Here is an example of such a recent result. It deals with the distribution of prime numbers, like 7, 11, 23, or 37, which you cannot divide by another natural number other than 1 or themselves. We've found prime numbers as large as 22m digits long, and researchers are still looking.

If you look at a table of numbers, the prime numbers seem to be almost randomly mixed into the non-prime ones. For a long time, we have been able to describe the typical characteristics of prime numbers. As it turns out, prime numbers slowly but steadily appear less frequently - they "thin out" - among the larger numbers. What's more, we can quantify this process in surprisingly precise terms.

With the average primes being further and further apart as we look at ever larger numbers, a typical question for maths researchers to ask is whether this process of thinning out also carries over to the smallest gaps between primes. In other words, will all large primes come at increasing distances from each other, or will we always find primes that are close to each other?

A breakthrough result in 2014 showed that no matter how high we go among the numbers, we will always find two primes that are closer to each other than some constant number. That number was initially a whopping 70m. This might not seem very close but the fact that we could identify a finite number was an important breakthrough. Other mathematicians then set out to reduce this value, and the best I am currently aware of is a much more manageable 246.

Real applications - eventually

You might wonder how solving such abstract problems helps anyone outside of mathematics. First, there is a trickle-down effect. A fundamental result is useful in obtaining other pure mathematical results, which in turn are used to develop applied mathematics, which are then used by non-mathematicians. Second - and more importantly - mathematical theory is often ahead of its time, and the abstract nonsense of yesterday underpins the applied mathematics of today.

For example, number theory is the area that examines, among other items, questions like our prime number example. For many years this was considered the ultimate pure mathematics topic and completely unusable for any purpose other than satisfying human curiosity. The eminent early 20th-century number theorist and pacifist, G.H Hardy, was very proud to say: "No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world." In other words, he was glad his work could not be used for military purposes.

Nowadays, the number theory results that seemed so useless less than a century ago are at the heart of the encryption algorithms that let us securely order a product or check our bank accounts online. In a way that would have horrified Hardy, British intelligence services had actually already discovered the same method in secret ahead of their civil colleagues.

When the next technological or scientific breakthrough requires a new type of mathematical model, it is likely that the subject already has the underlying theory in hand, waiting to be adapted to a new setting.

Underlying all of this is one of the fundamental truths about mathematical research. The applications of mathematics might change with scientific progress, making some mathematical topics more useful at times than others. But because mathematical results are based on logical deductions alone, they actually never become wrong, never get obsolete, and never truly get old. They are just waiting for the right application to arrive.

This article was originally published on The Conversation. Read the original article.