Fifteen in a jiffy, take the escalator

The name itself is interesting. There are many famous theorems in mathematics: Pythagoras theorem, Gödel’s incompleteness theorem, Fermat’s last theorem and more. There are others that have a number in their name – the four square theorem, the four color theorem… but is one named for a number? I do not think so.

Because the name is so intriguing, this column will try to figure out the fifteen theorem.

It was first stated in 1993 by the late greats John Conway and WA Schneberger, and here it is:

The Fifteen Theorem: If a positive-definite quadratic form with an integer matrix represents every positive integer up to 15, then it represents every positive integer.

I spent a few minutes staring at those words. In addition to the expected mention of 15, he reminded me of the elegant proof technique of induction, which I first introduced in college-level mathematics. In short: if what you want to prove is true for a particular “base case”, and if you can move on from there to the next case (“inductive step”), then it completes the proof.

If this sounds transcendent at best, consider the two examples as metaphors.

First, climb a flight of stairs. How do you convince me that I can board the whole flight? First, by showing that I can set foot on the first ladder. Second, by showing that from there I can climb another ladder. That’s all I need. Because now that I’m sitting on the second ladder, I use the same ladder to reach the third, then the fourth, fifth and so on… and suddenly I’m at the top.

Second, breaking a bar of chocolate. Let’s say the bar is divided into 20 smaller squares by the usual shallow lines. How many breaks do I need to get 20 individual pieces? Well, start with a bar that has a square. Zero brake needed. two classes? Obviously, just a break. three classes? A break gives you a different piece and a two-square bar, and the latter requires another break. So a three-square bar requires two breaks – and so on. With each break, we reduce the bar to the first two cases (such as the single piece and two-square bar) whose breaks we know count. Thus we understand: a 20-square bar requires 19 breaks.

Metaphors like this help to illustrate what a proof by induction really is. Again, this requires two components: a “base case” (legs set on step #1, a one-square chocolate bar) and a “motivational stage” (climb to the next step, breaking off the chocolate square). In mathematics, induction is a powerful tool in the proof of myriad theorems. In the statement of the fifteen theorem, there is a hint of induction to me. What this says is that if we can prove something complex (whatever it is) for integers up to 15 – the base case is – then we have proved it for all integers.

But what is that complicated thing?

Let me try to start with the four-square theorem, which made an appearance above. It dates back to 1770, when Joseph-Louis Lagrange proved to be a really tempting proposition. He showed that any natural number (that is, 0 and positive integers 1, 2, 3 and so on) can be expressed as the sum of more than four squares of the natural numbers (the numbers 0, 1, 4, 9). ) , 16 and so on).

Try it for yourself:

0 = 0

45 = 36 + 9

69 = 64 + 4 + 1

142 = 121 + 16 + 4 + 1

e.t.c.

In algebraic terms, Lagrange proved that any natural number N can be written as:

n = a 2 + b 2 + c 2 + d 2

In which a, b, c and d are also natural numbers. Taking the fourth example above, n is 142; A is 11, B is 4, C is 2 and D is 1. Since this is true for any natural number n, mathematicians say that this equation is “universal”.

Impressed by the number of us humans, we have been grappling with such manipulations for a very long time. For example, why only the sum of four squares? What if we multiply any one of them by, say, 3; second to 5; And a third by 14? In other words, what if we give the coefficients a, b, c and d? In this case, if we wrote this:

n = a2 + 14b2 + 3c2 + 5d2

Would this still be valid for any natural numbers N, a, b, c and d – that is, is this equation also universal? Well, number theorist genius Srinivasa Ramanujan thought a lot about such questions. It should come as no surprise that he proved that there are 55 equations that are universal, although he was later shown to be wrong about one of them. As it turns out, the one above isn’t on his list. But there are many others, such as:

n = a2 + 12b2 + 4c2 + 2d2

n = 7a2 + b2 + 2c2 + 3d2

We’re getting closer to theorem fifteen, I promise you. John Conway and his students worked on such equations in 1993 when he was teaching at Princeton University. Not only Ramanujan’s 54, but all equations that use any coordinates for a, b, c and d: were they all universal? My intuition is that Conway had induction in mind, because he thought, why not check some values ​​for n? If we can show that any such equation can generate those values ​​- a base case, if you wish – perhaps that would lead to a proof of universality.

Conway was right. They were able to show that if the equation produces numbers up to 15 (1, 2, 3, 4 … 15), then it is universal.

Here’s something that almost leaves me speechless. Think about it. Take a random equation like the one above:

n = 131a2 + 3b2 + 16c2 + 8d2

Conway proved that if it could produce 1, 2, 3 … all the way up to 15 (just 15!), it would also produce 971,402 and 7,444,333,131,229 and, in fact, every other natural number as well.

Now you know why it is called the fifteen theorem. You may also have thought that “the positive-definite quadratic form containing the integer matrix” – derived from the statement of the theorem above – is just math-dialect for these equations we are discussing. The proof that Conway and Princeton students found was complicated and they never published it. But in 2000, and fittingly because it came from Ramanujan, the stellar Canadian mathematician of Indian origin Manjul Bhargava found a much simpler proof (see Fig. page 27 ,

This may be simple for mathematicians, but it is far beyond my mathematical ken. Still, there is a sweet reminder of the metaphor used above regarding stairs. Bhargava does not use stairs, but a numerical tool he calls “escalator lattice”. He talks of the two-dimensional escalator being “ascended” to three dimensions, then four, and finally five dimensions. Ignore what all this means. But “with a bit of fear,” Bhargava writes, “we may again ask whether any of these five-dimensional escalators proceed.” That is, are we going to move on forever?

“Luckily, the answer is no,” Bhargava tells us. “All five-dimensional escalators are universal.”

And he proves fifteen theorems. We are at the top of the ladder. Have some chocolate.