The Sums of Squares are Closed under Multiplication

Take the squares of any two integers and add them together. By such a process it is possible to get an infinite set of outcomes, {0, 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20... etc}. Now if you multiply any two of them together you get another number that is in the set; the sums of two squares is closed under the mathematical operation of multiplication.

This property of numbers seems first to have been recorded by an Indian mathematician and astronomer, Brahmagupta, in the 7th century. For this reason it is often called Brahmagupta's identity, but in 1225, the well-traveled Italian mathematician, Leonardo of Pisa, included the idea in his "Liber Quadratorum". Leonardo is often now called Fibonacci, and his books became so influential that the identity was often named for him. It is easy to show the identity is true with simple high school algebra.

Let a, b, c, and d be the four intergers to begin with, then we need to show that (a^{2}+b^{2})(c^{2}+d^{2})= x^{2}+y^{2} for some x and y.

Brahmagupta and Leonardo showed that x can be written as (ac+bd) and y can be written as (ad-bc). If you square each of these and add them together, you get (a^{2}+b^{2})(c^{2}+d^{2}).