I'm sorry to disappoint you, but your findings will be of little use to number theorists or crypologists.
There's a very simple explanation for why the primes are lined up on special rays like that, and it has to do with modular arithmetic.
24 has 2 prime factors: 2 and 3.
All the non-prime rays contain only multiples of 2 or 3. This is also why 2 and 3 don't fit your pattern. 24 is a nice number because it has lots of
little factors, thus the prime lines are going to be consolidated and nice-looking. But you could do the same thing with just about any number. For
instance, if we used 20, then the prime lines would be: 1, 3, 7, 9, 11, 13, 17 and 19, also forming a symmetrical pattern. 20 has 2 prime factors, 2
and 5, so all I did was take out all the rays divisible by either of those. In general you would do the same with any modulus number (size of your
circle), and unless your circle itself was prime, you'd get some rays with primes and some without.
The idea of using this to get primes has been known for a long time, and although you display it differently, it's really no different than a prime
Finding all the squares of primes (except 4 and 9) on line 1 is really cool, something I didn't know, and I salute you for it, but there's a reason
for this, too. Modular arithmetic says we only need to check the squares of the first circle. If they all lie on line 1, then all possible prime
squares will lie on line 1. Now, all exponents of primes will only lie on prime lines (except exponents of 2 and 3), so we can rule out 16 of the
lines already. Of the 5 primes we have to check, their squares happen to lie on line 1. That may seem like a big coincidence, but there's often some
interplay between your numbers and your modular base that make it statistically more likely. As an analogy, I once looked at numbers that can be
written in two different ways as the sum of two squares. For example, 50 = 1 + 49 = 25 + 25, so 50 is the first such number. It turns out that these
numbers are virtually guaranteed to be divisible by 5. This baffled me until i realized it had to do with how the first few squares line up in a mod 5
There's also a good reason, not just convention, why 1 is not a prime like you say it should be. An equivalent way of defining the primes is: "no
prime is a multiple of another prime." But this breaks down if we include 1, since every integer is a multiple of 1.
Your equation will not be of use in factoring large numbers. You may have noticed that the equation you use factors into:
p1^2 * (p1^2 + 24n - p2^2) = 0
When you factor it you can see that the roots of the parabola are at p1 because you've defined it that way. But you've just shuffled off the hard
work of factoring from yourself to whatever algorithm the graph display program uses. You couldn't use this to factor big (300 digit) primes without
taking far more time to display the graph than factoring it a more normal way.
Most number theorists think that the best way toward better understanding primes is through something called the Riemann Zeta Function.