ON NUMBERS EQUAL TO THE SUM OF TWO SQUARES IN MORE THAN ONE WAY

Leonhard Euler gives no proof to the statement r=(aa=bb)(cc+bb). Since none of the authors were aware of this idea, Li et al tried to prove it, and found an advance proof using the unique factorization property of Gaussian integers. This proof had the disappointing feature that it did not show how to find the numbers a, b, c, and d, but simply demonstrated their existence, Later, they found a constructive proof and they surprised to see it how it used mainly precalculus mathematics with some number theory. Here, Li et al show this derivation hoping that it will be of interest to serious students of mathematics at all levels.