Well as you might have known from testing the math work yourself, there are many fundamental problems with my Prime number solution. I did not mind that the computer program is always true, because that is not what I based my theory on. The fact that a spiral with radius = [(Prime #) / Pi] and arc length [S = to the Prime number in the series], is just that, “a spiral.”
As it appears to me now having drawn the spiral for the first Prime numbers under 100, the spiral formed is irregular instead of increasing at a steady value as a logarithmic spiral should. At this time I have no equation (other that s = r * theta) to describe it.
It may be of use to note that any number divided by Pi is in increments of 0.318471. So if you were to take 43 for example 43 * 0.318471 = 13.6943 = 43 / Pi. This creates a nice involute, but it describes any number, not just Prime numbers. However if the Prime spiral with radius = [(Prime #) / Pi] had an known equation it could be set to equal that involute and where they meet would be the Prime numbers.
Maybe I should have concentrated with a logarithmic spiral of the rate of change. Or investigate a log spiral of the rate of change, of the rate of change of the Prime numbers. I still have a “boxed” log spiral that has a [chord of the Prime number / Pi]. In it the spiral seems to fit better instead of being fixed in increments of Pi. There is something to investigate there.
I received an anonymous email that showed the equation part is always true. It is always good to receive feedback if only to know if you are right or wrong. Here is the work:
P2 / [((P2-P1) / Pi) + (P1/Pi)] = Pi
=> P2 / [(P2-P1+P1) / Pi)] = Pi
=> P2 / [P2/Pi] = Pi
=> 1/ (1/Pi) = Pi
=> Pi = Pi
How did I make such a mistake and why didn’t I check my work? The fact is I put all my effort in solving the logarithmic spiral. I don’t mind so much as such a simple equation could solve a 3000 year old problem. My work only partly depended upon it. I know it is naive to think you can solve such a problem in one night. However, I had this envision of the logarithmic spiral solving all kinds of series. I should have tried a known series first! However the main part of my work (click here for spiral_encryption and here for function_finder) did not rely on this equation working. Instead it relied on the truthfulness of the logarithmic spiral.
This problem isn’t finished, but it will be a lesser priority. I have put a lot of effort and work into it. It would be a waste of good (though mostly not correct) work. However as a book of unsolved problems says: “It does not recommend any of these unsolved problems... since the greatest mathematicians have been unable to solve them.”
Even when you don’t solve the math problem you learn something. I’ve studied cryptography, Prime theory, Laplace transforms, and geometry. I had a lot of theories most have proved untrue. But what is the established ratio of good to bad ideas? You know, the one were you have so many bad ideas until a good one is found.
You can read the discussion here at http://www.3dbuzz.com/vbforum/showthread.php?t=157642
I am not finished with this problem. I believe there is more to the story of the spiral that is envisioned as a boxed logarithmic spiral. (You can find its picture here in the Prime Summary.) As with any good math problem there is a challenge to it. I will still be updating the site with work as well as other projects. So until then...
May the Creative Force be with You
Hopefully it is clear what is being attempted to be solved here. I will post updates to better explain and hopefully solve this problem. This is a good group project. If you have read this and want to work on a problem email: email@example.com . Also more math can be found in the math_hunches section of Constructor’s Corner.