Factoring using waveforms

So, without mincing words, the approach I originally proposed is certainly incomplete, however I was unknowingly  stumbling into some interesting concepts.

Here's the core of what I found out through this: 

All factors of numbers are harmonic with the composites made of them.
When graphed in XY mode on an Oscilloscope, a number and its harmonics will not rotate, regardless of relative phase between them
Wave-wise, This rotation is also akin to the beats one would hear if two notes were out of tune

What I was working with was part of the baby steps needed to understand shor's algorithm.  The core concept that all numbers can be represented as frequencies, constructive and destructive interference, as well as other important things.

Something that I learned later:

Shor's algorithm operates based on periodicity.  For a given semi-prime, If we take the length of the periodic cycle of ((2^X)mod N), where x is the step number, we can learn about the P and Q that N is made of (assuming N is an integer not divisible by 2).  Specifically, (P-1)*(Q-1) = the length of the periodic cycle of ((2^X)mod N) * an integer.

To put it another way, if the length of that periodic cycle is known as W, we can say ((P-1)*(Q-1))mod w = 0.  Which is really pretty cool seeing as right there we now know that any given factor of N has to fit this pattern and has to be between 3 and N/3.  Factoring the number becomes more like tuning a guitar than doing hard math.

It doesn't sound like much, but by using some cool tricks with phase analysis on a quantum machine (assuming there's one capable of doing so), we can actually get results.  Without a quantum machine, you can (and I did) actually even use this information to factor things on paper.  Note: for doing it on paper, unless you hate your life, don't go above 4 digit semi-primes.  Even a 4 digit semi required the use of wolfram as my tiny calculator did not appreciate trying to modulus through a periodic sequence of 48.

At this point, I'm fairly sure this approach is naive. There is a merit in the realm of analyzing periodicity and in the fact that audacity, from my understanding, works on the basis of a FFT of the wave itself. But as stated before, a shot in the dark was all it was, it was a miss, but hey, it was an interesting thought. When I learn more about signal analysis I might take another look at this concept, but until then, I'll put a fork in this one.

EDIT2: Tomorrow is a very long time away sometimes. Spoke to a higher up professor in physics, specifically, one whom specializes in acoustics and waveform analysis and while we can see that I was getting results, by all means, I shouldn't be. So the next step is to refine the test to better account for error. The new wave output will be generated via mathmatica and the analysis done by Raven, a software from Cornell. The next step after that will be dependent on what the new results are. If they are in fact error caused by poor fidelity, then I'd like to try frequency modulation and a few other things. Moreover, if it is an error, but it still gets accurate results, then it would be worth looking into what can be done to replicate that error to turn it into an intended feature.

Past that, it's been a matter of learning more, increasingly difficult maths. Chipping away at pdf after pdf in order to understand concepts. The newest document I'm reading over has a strikingly similar concept behind it. It essentially boils down to the fact that division is a very costly operation and by performing said division on a physical system (for example, a wave) it could speed up computation time, even for something like factorization.

Anyways, back to studies


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EDIT: Got distracted by video games so it will wait until the morning
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I had a really nicely worded post and firefox decided to crash. I plan to rewrite it again in the next few minutes, but until then, here's a placeholder to remind me of content included until I edit this post back to its former state.

Added reference links, have a lot more to add over the next week or so
Spending a few hours a day researching
Lots of crazy subject matter intersecting
Post some equations I figured out

Been testing a few things out and found some materials that seem to suggest that I may be on the right line of thought. Grabbing one of the physics professors at my university to see if he can give me advice and either confirm or deny that what I'm suggesting is in fact possible.

To restate, Unlike most approaches to factoring, my goal is not to get exact numbers but instead to use the frequency sweep to quickly narrow down the number of possible factors from N to a much smaller fraction of N which can then be computed via standard means. Like tuning the top string of a guitar by playing the bottom string and listening to the wobbles (almost exactly actually).

http://arxiv.org/pdf/quant-ph/0503228v1.pdf

This link is one of the papers I've been looking through that seems to confirm my general thoughts on the matter, but I intend to get a more educated opinion before I commit to this as my graduation project.

Emails will be sent out today, meeting with the profs should occur in a week or so depending on how busy they are.