Friday, 10 February 2012

The four-square theorem

Another math day, while in Kitchener ...

In 1770, Lagrange proved that any prime number p can be written as a sum of four (integer) squares:


By making use of this fact and Euler's previous discovery (see this previous post of mine) that the product of sums of four squares can be written as a sum of four squares, Lagrange has now proved that every positive integer can be written as a sum of four squares.  For example,

2012 = 44^2 +6^2 +6^2 +2^2

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