Harristotelian Logic

As a physics undergrad I remember being taught that the "hidden variables" idea for physics was not considered to be valuable though it had its proponents, and now many years later I'm fascinated by what might otherwise be considered "pure math", where it has me re-thinking my physics on that question as well.

Simply enough I found that you can solve for f_1 and f_2 when f_1*f_2 = nT, where T is some target composite to be factored and n is a non-zero integer control variable, modulo some prime number p, where

f_1 = ak mod p and f_2 = a^{-1}(1 + a^2)k mod p

and 'a' is related to k mod p by

a^2 = (nT)(k^2)^{-1} - 1 mod p.

And it is a remarkable case where what could be considered previously hidden variables are brought to light.

For those wondering if those equations are valid or not, or simply the musings some deluded crackpot, I can assure you they do work, and getting them to factor is relatively simple, as you pick a prime p greater than sqrt(T), and pick a positive integer k, as close to p as you can get, like k=p-1, and then try to solve for 'a', and if you can, and also ak is greater than p, then you will have factored nT mod p, with high likelihood you factored nT itself.

Here's an example to demonstrate and then I'd like to talk about how the solution is derived:

Example: Let T=119. Then p=11 is greater than sqrt(119), and trying k=10, with n=1 gives

a^2 = (119)(10^2)^{-1} - 1 mod 11 = 8 mod 11,

but 8 is not a quadratic residue modulo 11, so no 'a' exists for this case. Trying now k=9, gives

a^2 = (119)(9^2)^{-1} - 1 mod 11 = 4 mod 11

so a=2 is a solution. And ak = 18, which is greater than 11, so

f_1 = ak = 18 mod 11 = 7 mod 11.

And you have a non-trivial factorization, as 7 is a factor of 119. Which can almost seem magical until you know the underlying mathematics, and also understand just how powerful the result is in terms of a high probability of success, as in fact, a k that will non-trivially factor can be shown to be in the neighborhood of k such that

abs(nT - (1 + a^2)k^2)

is a minimum, and I call that value k_0. Those needing further convincing before continuing can simply try more examples of their own, as now I'd like to talk about how these equations result and then return again to the ramifications for physics theory.

The equations come from an analytical study of the relatively simple quadratic system:

x^2 = y^2 + S

and

z^2 = y^2 + nT

where traditionally mathematicians have approached factoring with methods based on the even simpler

x^2 = y^2 mod N

where N is the composite to be factored.

I went to a more complex system because I was questioning whether or not you could factor one number through factoring another which I designated surrogate factoring, so 'S' is the surrogate to be factored, where you find z from finding x and y with the prior equation, in my initial probes of the efficacy of this approach.

However with time—the final result is the distillation of over 4 years of basic research—I found that instead of factoring S itself, I could force a prime p of my choice into S—forcing a factor—which gave me the system:

x^2 = y^2 mod p

and

z^2 = y^2 + nT

and then simply by adding a few constraints I was able to derive the equations above. Those constraints are:

1. z = x + ak mod p


2. 2az = k mod p.

And that's all you need!!! It's fairly trivial from that simple set of equations to derive the solution I give at the top of this post, and it's also very simple to derive the result that shows where solutions can be found, so from the mathematical standpoint the result is locked in very basic and simple mathematical proof.

But its ramifications are huge, as clearly there are variables that are hidden unless you look for them by asking the right questions! Where here they powerfully guide integer factorization itself where factorizations are connected to each other in this remarkable way not known until you derive a solution from these otherwise hidden variables.

If the same can be true for physics systems then what questions are asked can decide what solutions can be found, as in this case with a purely mathematical system, I simply asked the question: can you factor one number through factoring another?

And found a system where previously hidden variables emerged where p is such a remarkable one that I call it a helper prime.

Imagine if, for instance, quantum chromodynamics could be simplified to the level of quantum electrodynamics simply by asking the right questions and finding systems dependent on previously hidden variables including helper ones like p.

The physical reality then might be that disparate groups of atomic nuclei could be interacting with each other on levels not previously imagined, which of course Einstein called "spooky action at a distance", like composites factor dependent on each other in a way not observable without the proper analytical approach!

It is my hope that I have made this research result relevant to the physics community as it is the community from which I was trained, and it is I think the best hope for a continuation of this line of thought and a proper evaluation of the possibilities as my efforts to garner appropriate interest from the mathematical and cryptological communities have been disappointing.

For the physics community the question of what hidden variables can look like in a simple system can be seen now, as well as a relevance between seemingly disparate elements.

Mathematicians for centuries have looked at factoring composites as individual accomplishments—number by number. But considering factorizations in pairs revealed a surprisingly simple solution for f_1 and f_2, when f_1*f_2 = nT.

What might we learn from our physical world if we simply ask questions in a similar way?

It is my hope that there will be people around the world driven by the need to know the answer.

Thank you for your time and attention.