Talk:Euler's factorization method

I have my own variation on the theme, which I shall demonstrate using the same numbers as in the worked example:

1000009 = 1000^2 + 3^2 = 972^2 + 235^2.

Pair off the squared numbers, odd with odd and even with even: {1000,972} and {235,3}.

Take one pair and put their half-sum and half-difference along the diagonal of a 2x2 square:

986 ===
===  14

Fill in the remaining spaces with the half-sum and half-difference from the other pair:

986  119
116   14

Now calculate the ratios reading across and down:

986/119 = 116/14 = 58/7
986/116 = 119/14 = 17/2

986  119      17
116   14       2
58    7

And the factors are:
58^2 + 7^2 = 3713
17^2 + 2^2 =  293