Using Diophantus' method find four square numbers...

• Jan 16th 2012, 09:46 PM
Brjakewa
Using Diophantus' method find four square numbers...
Here is a problem I am given: Using Diophantus' method, find four square numbers such that their sum added to the sum of their sides is 73. Why does Diophantus' method always work?

or alternatively, another problem is:

Using Diophantus' method, show that 73 can be decomposed into the sum of 2 squares in two different ways. Show that Diophantus' method always works.

Here is an example of Diophantus' method, on which I have questions:

Example:

To find four square numbers such that their sum added to the sum of their sides makes a given number.
Given number 12.
Now x^2 + x + ¼ = a square.
Therefore the sum of four squares + the sum of their sides + 1 = sum of four other squares = 13, by hypothesis.
Therefore we have to divide 13 into four squares; then if we subtract ½ from each of their sides, we shall have the sides of the required squares.
Now 13 = 4 + 9 = (64/25 + 36/25) + (144/25 + 81/25),
and the sides of the required squares are 11/10, 7/10, 19/10, 13/10, and the squares themselves being 121/100, 49/100, 361/100, 169/100.

Where does the x^2+x+1/4 come from?

What does subtracting 1/2 do, where is this evident in the problem?

How would I or anyone else get from (64/25 + 36/25) + (144/25 + 81/25) to 11/10,7/10,19/10, and 13/10??? This is my biggest question of them all. I would actually like to understand how to do this problem, so some explanation and an answer as well would be great so I can check the work myself, and be sure I'm correct as well.