Vijay Pasupathinathan

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**The 290 theorem**

Manjul Bhargava is another of this year's Fields medalists. He works on

**number theory**, which in its simplest form is the study of integers:

..., -3, -2, -1, 0, 1, 2, 3, ...

So when I say 'number' in this post, I'll always mean one of these!

When Bhargava was a grad student at Princeton, he read a book on number theory by the famous mathematician Gauss. Gauss was interested in

**quadratic forms**, which are things like this:

x² + 3xy + y²

or this

-3x² + y² + 4xz + yz - 7z²

Gauss was mainly interested in quadratic forms with two variables, but it's also fun to think about more variables.

I can hand you a quadratic form and ask:

*what numbers can you get if you plug in any numbers you want for the variables?*

Start with something really easy. For this one

x²

you can only get the

**perfect squares**

0, 1, 4, 9, 16, ...

But what about this one?

x² + y²

Can you find numbers x and y that make x² + y² = 100? How about x² + y² = 99? Remember, I'm using 'numbers' to mean numbers like these:

..., -3, -2, -1, 0, 1, 2, 3, ...

And what about this quadratic form?

w² + x² + y² + z²

It's a famous fact that for this one, you can get

*any*positive number by plugging in numbers for w, x, y and z.

What about this?

x² + y² + z²

Now you

*can't*get every positive number. Do you see why?

We say a quadratic form is

**positive definite**if whenever you plug numbers into it, you get something positive - unless all those numbers were zero. For example,

x² + y² + z²

is positive definite, but

x² + y² - z²

is not.

Okay, now you're ready. Here's something amazing that Manjul Bhargava proved with +Jonathan Hanke in 2005.

Here's how to tell if you can get

*every*positive number by plugging in numbers for the variables in a positive definite quadratic form. It's enough to check that you can get every number from 1 to 290.

In fact, it's enough to get these numbers:

1, 2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 34, 35, 37, 42, 58, 93, 110, 145, 203, 290.

Weird!

This is just one of many things Bhargava has done. Most are a bit harder to explain, but I described one here:

https://plus.google.com/u/0/117663015413546257905/posts/VouaWQnthn5

It's about 'elliptic curves', another really popular topic in number theory.

And in fact, the 290 theorem I just explained is secretly about elliptic curves! As usual in number theory, the

*statement*of a theorem may sound simple, cute, and pointless... but the

*proof*reveals a very different world, and that's what really matters.

Here's a nice explanation of the proof:

• Yong Suk Moon, Universal quadratic forms and the 15-theorem and 290-theorem, https://math.stanford.edu/theses/moon.pdf.

The original paper is here:

• Manjul Bhargava and Jonathan Hanke, Universal quadratic forms and the 290-Theorem, to appear in

*Inventiones Mathematicae*, http://www.wordpress.jonhanke.com/wp-content/uploads/2011/09/290-Theorem-preprint.pdf

There's a lot left to do. For example, Jonathan Rouse tried to show that a positive definite quadratic form gives all

*odd*positive numbers if gives the odd numbers from 1 up to 451... but he only succeeded in showing this

*assuming*something called the Generalized Riemann Hypothesis! Proving this is an extremely hard problem in its own right.

• Jonathan Rouse, Quadratic forms representing all odd positive integers, http://arxiv.org/abs/1111.0979.

#spnetwork arxiv:1111.0979 #fieldsmedal #numbertheory #spnetwork

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