Thursday, May 18, 2006

Classifying Quadratic Forms

This is probably not news for many of you, but I recently read the very well-written piece All Square by Ivars Peterson. In that piece, the author describes some results in number theory by Manjul Bhargava (Princeton) and Jonathan P. Hanke (Duke) on universal quadratic forms whose statements even I could understand.

A quadratic form is a polynomial with integer coefficients in which each term has a variable with exponent 2 or is a multiple of a product of two variables. Such a form is universal if it can generate all of the positive integers. Can one check whether a quadratic form is universal? The two mathematicians mentioned above have shown that it can, by proving that a quadratic formula is universal iff it can generate 29 different positive integers in the range 1 to 290! They also enumerated all of the 6436 universal quadratic forms in four variables.

Computer scientists like us will be happy to know that Jonathan P. Hanke uses computers in his work. (Look at this page.)

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