Richey Numbers- Solved at Last

A Quick Reminder: Richey Numbers are positive integers that can be written as the sum of two squares in exactly two ways, that is

N = a^2+b^2 = c^2+d^2

for a or be not equal to c or d.

I concluded that any number of the form (4m+1)(4k+1) is a Richey Number for m,k>0, 4m+1 and 4k+1 both prime. One of my earlier ill-formed conjectures that held me back, the assumption that any number of the form 5^h for h>1 is a Richey Number, proved to be wrong when h=5. It still poses a problem though- 125 and 625 don't fit into the rule I came up with.

In any event, the formula stated above accounts for almost all the Richey Numbers; almost every Richey Number is of that form, and almost every number of that form is a Richey Number. NOTE: This refers only to "pure" Richey Numbers; non-pure Richey Numbers arise when you take a "pure" one, one whose prime factorization is (4m+1)(4k+1) for 4m+1 and 4k+1 both prime, and multiply it by 2 or by some square number. These still have the properties of a Richey Number, but are simply multiples of "pure" Richey Numbers. For example, 130 is a non-pure Richey Number: it is (2)(65) = (2)(5)(13).

My next problem looks at a very similar kind of equation. I haven't determined a name for it yet, but here's the problem:

Find solutions in positive integers a,b,c, and d such that

a^2 = b^2 + c^2 + d^2.

This is essentially an extension of Pythagorean triples- I figured that eventually I could generalize the problem to one of this form:

Sum[i=0, k]{a[i]^2} = x^2

that looks for solutions in x, a[i] integers.