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All Square

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

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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.

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