What do number theorists do and want in life? #1

Obviously, this title is clickbait. The question, as put forth here, is not well-posed. It is not clear if an answer to this question exists at all, and even if it does, it is certainly not unique. Unlike how mathematicians are mostly in straitjackets in pop culture, we actually come in all shapes and sizes. Let me reframe the question then: What do I, as a number theorist, do and want in life? The first part of this question is relatively easy to answer, and I will do so in the first couple of posts in this series. The second question, though, is rather open-ended, and one might argue that it is still not well-posed. I don't want this to be an excuse for waffling, so, instead, I will choose to interpret it as what I want from my professional life. Since we are on LinkedIn, I suppose you should have seen this coming from a mile away.

This article is in three parts, each of which will be a separate post:

1. What do I do?
2. Cheating in mathematics
3. What do I want?

What do I do?

I investigate patterns in numbers for a living. This might seem rather self-indulgent and sometimes it is, but mostly, it is not. Cryptography, famously, is a by-product of such investigations. To a number theorist, though, the mathematics underpinning the modern cryptographic protocols is old news. The RSA cryptosystem, which is still widely used, goes back to the 1970s but the problem of integer factorisation it’s based on goes back at least to Eratosthenes, 2500 years ago. This is not to say that research in cryptography is stale, but it can’t be the only poster child of number theory, a branch of mathematics that was described by Gauss as the queen of mathematics. I don’t do number theory because cryptography exists. I do number theory because it deals with primes and in mathematics, only primes are real; everything else is a social construct. Though I admit that last bit was quite wanky, let me have this because this is as close to a mic drop as I am going to get in my line of work. My research is mostly concerned with finding and proving non-trivial patterns in the seemingly random world of (prime) numbers. It involves a lot of staring at the blackboard, mostly because many in the field are yet to adopt whiteboards, and personally, I like to watch wet chalk dry. It is less time intensive than watching paint dry and the excitement associated with ghostly symbols apparating minutes after being drawn is real. Simple atavistic pleasures of life. But there are also more avant-garde pleasures to be had in number theory. I do make use of some serious computational power in my research. NCI Australia’s shiny new supercomputer, Gadi, lets me (partially) verify number theory conjectures in hours rather than the months it would have taken on my friendly household PC, which goes by the name Desk Iguana. The reasons for it being named thus are myriad but suffice it to say that I think of it as a silicon-based reptile.

While silicon-based reptiles might well breathe mathematics, I am just a lowly carbon-based reptile. I mostly breathe the fumes from the wreckage of another of my attempts at solving a problem. That is the nature of research most days. Even on days when something does work out, the closest I come to Archimedes is when I take a long shower to reflect, in quiet satisfaction, on the arduous process I went through. Research in number theory and more broadly, research in mathematics, is often about chipping away at a problem even when it seems overwhelming at first. To solve a problem, you first need to read some of the existing mathematical literature and approach the problem with these acquired tools. If these tools don’t suffice, you read more and try to scale the problem again. This process is repeated until the problem is solved or it becomes clear that the problem is intractable using currently available technology. At this point in time, depending upon what you want out of a problem in terms of your career, you either decide to get your head down and develop the necessary tools on your own or you hold your head high knowing that you tried what you could but because of your current goals, you don’t want to spend more time on the problem. And boy, there are some rabbit holes out there.

If you think the mystery of Jack the Ripper is enduring, try the Goldbach conjecture on for size. I have often waxed lyrical about this problem at parties. My exploits in this area include convincing at least one philosopher to study mathematics more deeply during the course of a birthday party. On a completely unrelated note, I have not interacted with this person again. I think they are studying mathematics every waking moment and have no time for human interaction. My persistence paid off. I am quite persistent when it comes to solving problems too, even when certain aspects of these problems might require some grunt work. The amount of grunt work depends on the area of mathematics you work in. It is not something that we like to brag about but the focus of our research group is explicit number theory which requires a fair bit of grunt work. Before grunt and explicit give you any NSFW ideas, I should clarify that explicit, here, means that we provide the numbers (computations) in number theory. The more exalted minds in the area stick to the theory part of number theory.

So, how do I go about putting the numbers where they belong, er, in number theory? This will be addressed in my next post where I will discuss the Goldbach conjecture and how it relates to my current research. More importantly, I will also give you tips about how to cheat in mathematics.