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SUM OF SQUARES

One interesting fact about squares of numbers is that a sum of two squares times a sum of two squares is always a sum of two squares.  In equation form, this is manifested as

                       (A2+ B2) (C2+ D2) = E2+ F2

The above can be shown to be true as follows:

                            E2 = (AD +BC)2

and

                      F2 = (AC -BD)2

This fact was known as early as 650 AD by the Indian mathematician Brahmagupta.  Fibonacci also noted this around 1200 AD. 

The above theorem holds for both real and complex numbers. 

It can also be shown that if an odd number is a sum of two squares, then it is of the form

                    4*k+1

However, the converse is not true.  Not every odd number of the form 4*k+1 is a sum of two squares, even if one of the squares is zero.  One example is 21.

Fermat, however, proved that every prime number of the form 4*k+1 is a sum of two squares.

Lagrange also proved that a sum of four squares times a sum of four squares is always a sum of four squares. 

 

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