Once more do I have Gaurish to thank for the day’s topic. (There’ll be two more chances this week, providing I keep my writing just enough ahead of deadline.) This one doesn’t touch category theory or topology.
I keep touching on group theory here. It’s a field that’s about what kinds of things can work like arithmetic does. A group is a set of things that you can add together. At least, you can do something that works like adding regular numbers together does. A ring is a set of things that you can add and multiply together.
There are many interesting rings. Here’s one. It’s called the Gaussian Integers. They’re made of numbers we can write as , where ‘a’ and ‘b’ are some integers. is what you figure, that number that multiplied by itself is -1. These aren’t the complex-valued numbers, you notice, because ‘a’ and ‘b’ are always integers. But you add them together the way you add complex-valued numbers together. That is, plus is the number . And you multiply them the way you multiply complex-valued numbers together. That is, times is the number .
We created something that has addition and multiplication. It picks up subtraction for free. It doesn’t have division. We can create rings that do, but this one won’t, any more than regular old integers have division. But we can ask what other normal-arithmetic-like stuff these Gaussian integers do have. For instance, can we factor numbers?
This isn’t an obvious one. No, we can’t expect to be able to divide one Gaussian integer by another. But we can’t expect to divide a regular old integer by another, not and get an integer out of it. That doesn’t mean we can’t factor them. It means we divide the regular old integers into a couple classes. There’s prime numbers. There’s composites. There’s the unit, the number 1. There’s zero. We know prime numbers; they’re 2, 3, 5, 7, and so on. Composite numbers are the ones you get by multiplying prime numbers together: 4, 6, 8, 9, 10, and so on. 1 and 0 are off on their own. Leave them there. We can’t divide any old integer by any old integer. But we can say an integer is equal to this string of prime numbers multiplied together. This gives us a handle by which we can prove a lot of interesting results.
We can do the same with Gaussian integers. We can divide them up into Gaussian primes, Gaussian composites, units, and zero. The words mean what they mean for regular old integers. A Gaussian composite can be factored into the multiples of Gaussian primes. Gaussian primes can’t be factored any further.
If we know what the prime numbers are for regular old integers we can tell whether something’s a Gaussian prime. Admittedly, knowing all the prime numbers is a challenge. But a Gaussian integer will be prime whenever a couple simple-to-test conditions are true. First is if ‘a’ and ‘b’ are both not zero, but is a prime number. So, for example, is a Gaussian prime.
You might ask, hey, would also be a Gaussian prime? That’s also got components that are integers, and the squares of them add up to a prime number (41). Well-spotted. Gaussian primes appear in quartets. If is a Gaussian prime, so is . And so are and .
There’s another group of Gaussian primes. These are the numbers where either ‘a’ or ‘b’ is zero. Then the other one is, if positive, three more than a whole multiple of four. If it’s negative, then it’s three less than a whole multiple of four. So ‘3’ is a Gaussian prime, as is -3, and as is and so is .
This has strange effects. Like, ‘3’ is a prime number in the regular old scheme of things. It’s also a Gaussian prime. But familiar other prime numbers like ‘2’ and ‘5’? Not anymore. Two is equal to ; both of those terms are Gaussian primes. Five is equal to . There are similar shocking results for 13. But, roughly, the world of composites and prime numbers translates into Gaussian composites and Gaussian primes. In this slightly exotic structure we have everything familiar about factoring numbers.
You might have some nagging thoughts. Like, sure, two is equal to . But isn’t it also equal to ? One of the important things about prime numbers is that every composite number is the product of a unique string of prime numbers. Do we have to give that up for Gaussian integers?
Good nag. But no; the doubt is coming about because you’ve forgotten the difference between “the positive integers” and “all the integers”. If we stick to positive whole numbers then, yeah, (say) ten is equal to two times five and no other combination of prime numbers. But suppose we have all the integers, positive and negative. Then ten is equal to either two times five or it’s equal to negative two times negative five. Or, better, it’s equal to negative one times two times negative one times five. Or suffix times any even number of negative ones.
Remember that bit about separating ‘one’ out from the world of primes and composites? That’s because the number one screws up these unique factorizations. You can always toss in extra factors of one, to taste, without changing the product of something. If we have positive and negative integers to use, then negative one does almost the same trick. We can toss in any even number of extra negative ones without changing the product. This is why we separate “units” out of the numbers. They’re not part of the prime factorization of any numbers.
For the Gaussian integers there are four units. 1 and -1, and . They are neither primes nor composites, and we don’t worry about how they would otherwise multiply the number of factorizations we get.
But let me close with a neat, easy-to-understand puzzle. It’s called the moat-crossing problem. In the regular old integers it’s this: imagine that the prime numbers are islands in a dangerous sea. You start on the number ‘2’. Imagine you have a board that can be set down and safely crossed, then picked up to be put down again. Could you get from the start and go off to safety, which is infinitely far away? If your board is some, fixed, finite length?
No, you can’t. The problem amounts to how big the gap between one prime number and the next largest prime number can be. It turns out there’s no limit to that. That is, you give me a number, as small or as large as you like. I can find some prime number that’s more than your number less than its successor. There are infinitely large gaps between prime numbers.
Gaussian primes, though? Since a Gaussian prime might have nearest neighbors in any direction? Nobody knows. We know there are arbitrarily large gaps. Pick a moat size; we can (eventually) find a Gaussian prime that’s at least that far away from its nearest neighbors. But this does not say whether it’s impossible to get from the smallest Gaussian primes — and its companions and on — infinitely far away. We know there’s a moat of width 6 separating the origin of things from infinity. We don’t know that there’s bigger ones.
You’re not going to solve this problem. Unless I have more brilliant readers than I know about; if I have ones who can solve this problem then I might be too intimidated to write anything more. But there is surely a pleasant pastime, maybe a charming game, to be made from this. Try finding the biggest possible moats around some set of Gaussian prime island.
Ellen Gethner, Stan Wagon, and Brian Wick’s A Stroll Through the Gaussian Primes describes this moat problem. It also sports some fine pictures of where the Gaussian primes are and what kinds of moats you can find. If you don’t follow the reasoning, you can still enjoy the illustrations.