100 percent proof

Proof is the bottom line for everyone

In the introduction to Numbercrunch (paperback out on 11th April, please pre-order now!!) I wrote that

Most people don’t know what a professional mathematician does all day. Perhaps they imagine that we are memorising harder and harder times tables (‘one 19,573 is 19,573, two 19,573s are 39,146’)

but then I didn’t really explain what we actually do, so I’d like to do that here.

Substack’s AI generator had a go at it: not bad, I’d say

Some of my job involves teaching students (which is great!). Another, and apparently increasing, part involves administration (which is … not always great). But the remaining part of the job involves research.

On some level this means “finding new maths”, and I think that often comes as a surprise to people. Surely there can’t be any new maths left to find? Humanity has been working at this stuff for thousands of years, there can’t still be new sums left to do?

In fact, you’d be surprised. There’s lots of things we don’t know, and lots of people working to figure them out. You can think of it as a bit like prospecting for gold. Some people will happily stand in the same stream for a long time, panning gold dust out of the water, and putting it together every year or so into a published paper on fairly similar problems.

However, at the same time, new goldfields are constantly being discovered. Every so often somebody makes a wild leap, suggests we start looking somewhere else, and shows off the nuggets they managed to extract from that unexpected place. This development can open up whole new areas of interest, and often hundreds of people will swarm to the same place and start hunting there too.

The analogy breaks down beyond that though, because often some of the most interesting maths happens at the boundaries between disciplines. Maybe this is a bit more like old-fashioned alchemy, where ideas from Field X and Field Y can be brought together, and only in combination do they become gold.

None of this really answers the question, though. What is it that mathematicians are trying to find? This can vary a bit between different parts of the subject: there are disciplines which are adjacent to physics or engineering for example where other standards will apply. But for pure mathematicians, at least, the game is really to prove a theorem.

And this is where the subject feels quite unique, in terms of the standards of proof we set ourselves. Consider for example so-called ‘perfect’ numbers. These are numbers whose factors (the things that divide into it, not including itself) add up to the number itself. For example 28 has factors 1, 2, 4, 7 and 14 - and these numbers add up to 28 exactly.

Now, there is a recipe to find perfect numbers, which was known to Euclid. You might notice that 7 is both prime and one less than a power of 2. There aren’t very many numbers like this - the first few are 3, 7, 31 and 127, and in total we only know fifty-one of them, though we are always looking for more (and you can help!). Such numbers are rare enough and interesting enough that we give them a name - they are called Mersenne primes. The trick is that if you have a Mersenne prime, you can find a perfect number, by multiplying itself by ‘itself plus one over 2’. So for example, we multiply 7 by (1+7)/2 = 4 to get 28, which is the perfect number I mentioned above.¹

This trick will always work: every Mersenne prime gives us a perfect number in the same way. For example, the Mersenne prime 3 gives us the perfect number 6². So in total we know fifty-one perfect numbers so far, and all of them are even. In fact, once Euler came along we knew there wasn’t any other way to find an even perfect number.

But that still leaves the question: could there be an odd perfect number? Despite extensive searching, nobody has found one so far. And it’s not like people haven’t been looking - thanks to some clever mathematical arguments, we know that if an odd perfect number exists, then it must have at least 1,500 digits.

It’s hard to convey just how astronomically big a number this is. The number of electrons in the entire universe probably has about 80 digits. The number of possible chess games has about 120 digits. If you’d checked every single electron or chess games for some property, you’d probably feel like you’d made a pretty serious effort. But that doesn’t come close - this 1,500 digit number is really exponentially³ bigger. In fact, this is like checking every possible way that a combination of a dozen simultaneous chess games could play out.

So, for most people, if you’ve checked all the odd numbers with up to 1,500 digits, and none of them are perfect, you’d probably feel safe in concluding that no such thing exists.

Mathematicians are not ‘most people’.

We would say that checking this many numbers doesn’t even scratch the surface. We know that sometimes the interesting things don’t happen until you get into the territory of really big numbers. In fact, whatever point we check up to, there would still be infinitely many odd numbers that we haven’t checked. So until someone comes up with a clever argument that takes care of all of them, as far as we are concerned the problem is still open, and the answer is unknown. This may feel like an absurd level of pedantry, but that’s just the way that mathematicians think.

This is just an example of course. The question of whether there is an odd perfect number is part of a branch of maths called number theory, and I realised long ago that I’m not smart enough to do that stuff. There are many problems in that area which are absurdly easy to state (“every even number bigger than 2 is a sum of two primes”) and absurdly difficult to prove.⁴

However, on some level, although I don’t work in number theory my job is still to prove theorems. Some of my collaborations don’t involve this focus on rigour, but many of the papers that I’m proudest of do exactly this. For example, our paper on the Shepp-Olkin monotonicity conjecture has the following abstract:

Consider tossing a collection of coins, each fair or biased towards heads, and take the distribution of the total number of heads that result. It is natural to suppose that this distribution should be ‘more random’ when each coin is fairer. In this paper, we prove a 40 year old conjecture of Shepp and Olkin, by showing that the Shannon entropy is monotonically increasing in this case

You don’t need to worry about the last sentence, but this is again a question about infinite possibilities. We showed that however many coins you have, and however biased towards heads they are, then making one coin fairer makes the outcome more random.

It's kind of neat when you are able to do something like that. But I’d been thinking on and off about that problem for something like 12 years when we figured it out. And of course, most of the time, that means you aren’t actually solving a problem. So, the answer to “what does a mathematician do all day” is that we write on lots of pieces of paper that go in the recycling bin, and that we write on a lot of blackboards that get erased. We read other peoples’ stuff and listen to their talks, and hope to spot good ideas that might be relevant to the particular problems that we care about. We drink coffee and try to carve out chunks of time away from other distractions to think about problems. But mostly, we hope that if we try enough things that don’t work then we’ll eventually find the thing that does, because once something is proved it stays true forever, and that’s a nice thought to have.

1. You can have fun convincing yourself that this recipe always works. Go on, get a piece of paper out!

2. 6 is perfect because the factors are 1,2 and 3, which add up to 6. The next Mersenne prime is 31 which gives 496. It’s maybe less obvious that the factors here, namely 1, 2, 4, 8, 16, 31, 62, 124 and 248 add up to 496, and seeing how those fit together might give you a hint as to how the general recipe works.

3. Yay!

4. This is called the Goldbach Conjecture. There’s a related problem called the weak Goldbach conjecture (“every odd number bigger than 5 is a sum of three primes”), which would be true if Goldbach is. My former colleague Harold Helfgott announced a proof of the weak conjecture over ten years ago, but I believe that the paper still hasn’t been published in a journal: the community can be pretty demanding in terms of checking other peoples’ proofs too!