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All Square

A surprising, far-reaching overhaul for theories about quadratic expressions

9:21am, March 7, 2006

Start with the square numbers 1, 4, 9, 16, 25, 36, and so on. Pick any other number and you can express it as a sum of squares. For example, 10 = 1 + 1 + 4 + 4 and 30 = 1 + 4 + 9 + 16. In 1770, French mathematician Joseph-Louis Lagrange proved that every positive integer is either a square itself or the sum of two, three, or four squares. No more than four squares, x2 + y2 + z2 + t2, are ever needed to express any number, no matter how large.

Given Lagrange's result, number theorists asked whether there are other such expressions, called quadratic forms, that also represent all positive integer

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