New Findings

I have cataloged all of the Richey Numbers between 1 and 1000, and have come to some new conclusions.

First, I noticed that given a solution, N = a^2 + b^2 = c^2 + d^2, 2N is also a solution, and with some intuition I realized that 2N = (a+b)^2 + (a-b)^2 = (c+d)^2 + (c-d)^2. The solution I worked with first was 65- I noticed that 130 was also a solution.

65 = 8^2 + 1^2 = 4^2 + 7^2
130 = 9^2 + 7^2 = 11^2 + 3^2

As you may notice, 9 is the sum of 8 and 1, 7 is the difference of 8 and 1, 11 is the sum of 7 and 4, and 3 is the difference of 7 and 4.

This result is easily proven.

N = a^2 + b^2
2N = (a+b)^2 + (a-b)^2
= a^2 + 2ab + b^2 + a^2 - 2ab + b^2
= 2a^2 + 2b^2
= 2N

The same also follows for the other part, c^2 + d^2.

The next thing I began experimenting with was the prime factorizations of Richey Numbers. I looked mainly at the numbers I thought to be "pure"; those that are not iterations of other Richey Numbers. For example- 130 is only 65*2, and the prime factorization gave that of 65 with a 2 in it as well. Other examples include any Richey Number multiplied by a square #, because of another property I discovered: Given a solution N = a^2 + b^2 = c^2 + d^2, (k^2)N = (ka)^2+(kb)^2 = (kc)^2 + (kd)^2, an easily verifiable result. Therefore, numbers like 200, 260 and 450 are reiterations: 50*4, 65*4, 50*9 respectively. This finding, coupled with the principle noted above, also led me to include Pythagorean triplets into my definition of Richey Numbers. (This basically means that I would allow a, b, c, or d to be zero, something I hadn't allowed before.) I realized that if a^2+b^2 = c^2 was also considered as a solution, that may explain the prime factorizations of some of the numbers I already came up with, such as 50 and 388. This is because their prime factorizations turned out to be 2*k^2, for some integer k- in these cases, 2*5^2 and 2*169^2 respectively. At first, I was puzzled as to how something like 25, whose components (3^2+4^2 = 5^2) include a zero (in terms of Richey Numbers) could lead to a legitimate Richey Number. I quickly realized that when multiplying 25 by 2 and using the rule I stated above, the sum of 5 and 0 and their difference are both 5- both non-zero numbers, thus yielding 50, a more recognizeable Richey Number.

This analysis left four different kinds of prime factorizations, and I list them here in order of frequency, most frequent to least (N is a given Richey Number):

1. N = 5*Pk, where Pk is some prime in a specific set of primes.
2. N = Pa*Pb, where Pa and Pb are primes of the same set as in 1.
3. N = Pk^2, where Pk is a given prime in the same set as in 1.
4. N = 5^x, given a whole x>1.

The "specific set of primes" I refer to here was found only by studying the prime factorizations. These pure Richey Numbers' prime factorizations used these primes only, and I believe that there is some general relationship all these primes share (though I haven't yet found it). Here's a list of the first few pure Richey Numbers, their prime factorizations, and the categories they fall into:

25 = 5^2 (4)
65 = 5*13 (1)
85 = 5*17 (1)
125 = 5^3 (4)
145 = 5*29 (1)
169 = 13^2 (3)
185 = 5*37 (1)
205 = 5*41 (1)
221 = 13*17 (2)
265 = 5*53 (1)
...
And the list of primes:

13,17,29,37,41,53,61,73,89,97,101,109,113,137,149,157,173,181,193,197...

I hope this will lead somewhere interesting, as this list of oddly spaced primes does seem, well, odd.