On 37

Today's a big day. Later this evening, in just a few hours, Papa and I will hop down to the local courier concern and send my boxes off their way. Today's the day where we must be absolutely certain: if it doesn't go today and it can't fit into the luggage than it's not going. I think it's all in order. Thanks to the kind Party the Third the luggage recently got bigger. I think that's set my mind easier, but I'm still highly anxious about things I know not what.

I'm packing my speakers off, too, so I thought I'd take today to transfer all the data I need to the shiny new laptop. While this transfer proceeds, I thought I'd take a moment out to tell you about 37.

37 is a remarkable integer, and my favourite one. It's prime, of course. Other than that it doesn't seem on the face any more or less remarkable than, say, 47. Another principal candidate for Favourite Integer is 26, which besides figuring in my birth date is the only number between a square (25) and a cube (27), (a proposition dating to Fermat).

You may think that there are so many integers it would be impossible to have a favourite one. This is not so. First, the Law of Small Numbers suggests that small numbers are the really staggeringly remarkable ones; they just get more boring as they get bigger. Paradoxically, of course, this makes it all the more interesting when the smallest example of something turns out to be 37; but 78,557 being the smallest Sierpinski number doesn't really make it more endearing. It's a delicate balance. Someone who thinks Graham's number or Skewe's number or something like that is the best integer is, I'm sorry to say, lacking in taste.

Yesterday I happened to mention over dinner somewhat provocatively that 37 is my favourite integer, which I shouldn't have done, because it was not a good time to explain why, since it's a rather long story that will require a detour through a lot of elementary algebraic number theory. But having stepped in it, I had to say why; and it's an interesting story, to me, anyway; so here we are. Don't let the words distract you from the music. If it gets too bad, just imagine Kosh is saying it. (This works especially well if you have questions. Query: What does this mean? Answer: Yes.) For the mathematicians in the audience, I will beg your forbearance with the simplifications, beginning with my second definition.

A (rational) integer is a counting number, 0, 1, 2, ..., or the negative of one. The set of such is labelled Z (for zahlen, German "number"). Prime integers, as we learn in grade school, are those which are divided only by themselves and 1. Otherwise a number is called composite. Every composite number can be expressed as a product of prime numbers. In fact this prime factorisation is unique.

There are other kinds of numbers, which are not integers, like the square root of two, say. We can do a kind of generalised arithmetic with these numbers, too. The set ("ring") Z[α] consists of all the things that look like a + bα + ... + cαⁿ, where a, b, ..., c, n are some integers. We can ask about what sorts of properties of integer arithmetic carry over to these new kinds of arithmetic. One question is: is there also unique factorisation into primes in a ring Z[α]?

It turns out that the answer is usually no. An easy example is to take α equal to the square root of -5. Now we can write 6 = 2*3 = (1 + α)*(1 - α), and it turns out that all of 2, 3, 1 + α, and 1 - α are irreducible in this ring, so that these factorisations are "essentially different." (Irreducible is related to but not quite the same as prime in a way which I will not say.)

However, there is a very deep theorem due to Dedekind on this subject. The first idea is to introduce so-called ideal numbers, not all of which exist in the ring Z[α], but which are related to the numbers in this ring. With these numbers, unique factorisation can be restored. This is an excellent achievement because unique factorisation is a very strong property and a lot of consequences follow from it purely formally. It turns out, to give you an idea that this is not so strange, that the only ideal numbers you need can be denoted (a), which kind we identify with just a, or (a, b), which we can think of as the greatest common divisor of a and b, for a, b elements of Z[α]. (All of this is true only for certain α. I won't say which, but all the α I mention in this post are of this kind. An example of something that doesn't work is π. Another example is the square root of 5. It's tricky.) There is an object called the class group which one can define from these ideal numbers. Very roughly speaking, it tells us how many essentially different kinds of factorisations (into non-ideal numbers) there are in the given ring. The very deep theorem of Dedekind which I mentioned is the statement that the class group is a finite set. Its size is usually denoted h (and depends on α, of course). We have unique factorisation if and only if h = 1.

With this behind us, we are going to specialise to the case where α = ζ_p is a (primitive, complex) p^th root of unity. This means that α^p = 1, but α is not equal to one. For example, the number i which solves the equation x² = -1 is a fourth root of unity. Now let h be the class number of the ring Z[ζ_p]. We say that the prime p is regular if p does not divide h, and irregular otherwise.

37 is the smallest irregular prime.

You may wonder why anyone would be so daft as to make this definition in the first place: why is this property of any interest at all? The answer, as it is with so many things in elementary algebraic number theory, is Fermat's Last Theorem. It is possible to give an "elementary" proof of FLT for the case where the exponent is a regular prime.

(For the math guys who haven't read it. The idea is to start with x^p + y^p = z^p and factor the LHS into a product of terms looking like x + ζ^ky where ζ = ζ_p is as before. Now if we had unique factorisation in Z[ζ], a product of relatively prime factors being a p^th power means that each factor is itself a p^th power, a very strong condition, which we can get our hands on by passing to ideals. For the other case we note that the gcd of two of those factors divides also their sum and difference. In any case, there is a long and difficult calculation ahead, which takes a few pages after the preparatory lemmas are stated. At a critical juncture we have some ideal I whose p^th power is principal. If p is regular, then it is coprime to h, and this implies from the definition of the class group as fractional ideals modulo principal ideals that I itself is principal. That's the only place where it matters that p is regular. I think there is also some assumption that p doesn't divide xyz in the argument I remember.)

Frankly, I find it remarkable enough that there are any irregular primes at all. There are three less than 100. It turns out that there are even infinitely many irregular primes. I guess when I understand this fact I won't find it so astonishing that there are any at all. But the other surprising thing is that we don't even know if there are infinitely many regular primes, although it's conjectured (and there seem to be more regular than irregular in the ranges where we know). You can take a look around some of the links on the right column to read some strange facts and more technical descriptions if you like.

Postscript, immediately after. That ζ looks really ugly in this font. Shame, it was a favourite Greek letter of mine. Also, tthere's a post from "last night" [early this morning] just below here, too.