Richey Numbers: An Overview

After reviewing this problem further for a while, I came to some new conclusions.

First, (I stole this one from a book by Paul J. Nahin), there is an alternate way to generate numbers of the form a^2+b^2 = c^2+d^2. His method uses the product of the sum of two squares:

(u^2+v^2)(x^2+y^2)

The theorem says that a product of this form (for integral values u, v, x and y) will always be equal to a number that can be expressed in two distinct ways as the sum of two squares. This can be verified by some algebra and the solutions can be found in terms of u, v, x and y.

This, along with a new ability I have maintained to use C++ code to find solutions to some of these number theory problems, led me to write a program that found solutions, and it works quite well. This leads me to another, quicker conclusion. The smallest Richey number that is not a multiple of 5 is 221, not 533 as I had said earlier. (221 = 5^2+14^2 = 10^2 + 11^2)

Finally, there is another element of these numbers that I have not yet studied. There appear some relationships between the numbers which compose my numbers. One example, which I noted last time, was the relationship that a+b-c-d was always an even number. There appear some more of these strange relationships, but I haven't yet determined anything with proof.