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.

Richey Numbers: Update

Recently, I discovered a strange list of primes while looking at the prime factorizations of Richey Numbers. (Dated January 3rd.) I the primes arose in the prime factorizations of Richey Numbers, and only those primes appeared in the factorizations. I realized that the list is all the primes of the form 4m+1, for m>0. I therefore conjectured that any number of the form 5(4m+1) = 20m+5 is a Richey Number when m is prime. (The 5 is there because the primes appeared in the factorizations attached always to a 5. For example, 65 = 5*13, 85 = 5*17, and so on.)

This finding also accounts for numbers of the form (4m+1)^2 = 16m^2+8m+1 and (4m+1)(4k+1), for 4k+1 and 4m+1 both prime. However, I haven't yet accounted for one strange kind of Richey Number; those of the form 5^n for n>2, such as 125, 625 and 3125.

I haven't algebraically shown why this is true yet, and I'm not really sure how to go about it. I need to somehow determine when 4m+1 is prime, and I'm pretty sure no one has done it yet. My guess is it's just as hard as determining when any given number is prime, certainly an extraordinarily difficult if not impossible task. I'll try looking at the components of the Richey Numbers of the form 20m+5 and see if there are any useful trends.