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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



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 Mathematicaehttp://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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