The Representation of Integers by Quadratic Forms

There's a nice article at Science News on the work of Bhargava on the representation of integers by quadratic forms. Rather than just restate what's written there (and in a few other blogs) let me quote the main theorem which is really quite amazing:

If Q is a quadratic form: Zⁿ->Z then if the image of Q contains {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}, then it contains every positiver integer.

Closely related is Conway's 15-theorem which originally inspired Bhargava and which (I think) I first read about in the excellent book The Sensual Quadratic Form Check out the section on topographs where Conway works his usual magic and makes some parts of the theory seem so clear and obvious that even your cat could start proving theorems about quadratic forms.