Number Theory and Physics.

[jwaltos]

The following belongs in this section since only thin thread of verifiable fact binds the content. I have partially
worked out a form of integer factorization where I need to utilize a better formalism to package this methodology so that
I may effectively broach certain questions in mathematical physics.

In Spivak, Vol.III p.173 is a Scherk minimal surface and in a paper by Szendroi on cubic curves, the latter
portion mentions superstrings as a fundamental building block rather than a point particle. The method
of integer factorization I have uses a periodic parabola as the fundamental building block much like Spivak's
figure. A catenary is generated via the focus of a rolling parabola. If a linear wave perpenicularly enters a
parabola an expanding circle is generated at the focus. Intersecting circles give rise to hyperbolas and ellipses.
The inversion of a parabola through its focus creates a cardioid and Mandelbrot's fractal form can be generated accordingly.
Consider such a 2 dimensional coordinate structure as a sheet referencing a specific residue and then
consider a manifold of such sheets such that every possible integer factorization has been accounted for.
Now consider each such sheet as a 3 dimensional structure where the parabolas are paraboloids and the circles
become spheres. Finally, consider that every geometrical object and path relates directly to a factorization
described as a numerical coordinate on a parabola.

The above parabolic basis is predicated upon Fermat's difference of squares. I have derived a general
form of this relation for integer factorization where the inner square is still a square value but the
outer shape is no longer a square and a more general form of integer factorization comes to light. Another
aspect of this relation is when the difference of cubes is considered and this 3 dimensional figure is
transformed into a torus. Consider how every operation conducted within this referential frame affects
the parabolic basis and conversely.

A paper on Weierstrass-Enneper Representations (Kilchrist & Packard) links minimal structures to Hamiltonian systems.
Relative to physics, a Riemannian manifold is defined in terms of quadratic forms which allows for the
construction of a metric space. Some of my reference texts and papers include Nambu's collected papers, a text
on the Dynamics of the Standard Model, Wienberg' Gravitation and Cosmology as well as papers on the Higg's
particle and the Higg's Field (easily obtainable online). Raamsdonk's papers associating entanglement with the
creation of space-time should relate back to the construction of a metric space and be consistent with
the mathematics of the Standard Model and General Relativity..but not completely.

So there it is, I'm grappling with aspects of number theory and coordinate geometry that seem to coincide with
certain aspects of mathematical physics and am trying to determine which is the most clear, consistent,concise and
intuitive formalism that will allow me to navigate the above as seamlessly as possible.

To conclude, the Wikipedia pages referencing the Dirac Equation and Freeman Dyson associate with some of the above content
.. and as an homage to this section I'll cite Dyson's "Crank...of a partition."

[Nick]

Are you sure that differential geometry is the best choice of geometry for what you are doing?

[jwaltos]

Regarding the integer factorization method I developed and which I believe is new, here are some equations, working
programs and preliminaries:
The syntax is Maple and the material dates from 2005, just copy the text into a Maple worksheet, make sure to remove
the leading ">" greater-than symbols and reformat certain execution blocks otherwise the program will not work as intended.
If you don't have Maple then you should be able to easily translate the text for your preferred software. The equations
and the program which generates equations from two factors are the end points of an integer factorization
spectrum. There exists a transformation which I call "progressive duality" that creates all of the intermediate
factorization forms within this spectrum that I am cleaning up for later inclusion but I have provided an
example.
The coordinates used: n1->references the location of the parabola along the x-axis (+ve to the right)
m1->references the location of the parabola along the y-axis (+ve downward)
[interchangeably] xa/xb->references a point on the parabola

The equation, V=L*27000+K: V->number to be factored
L-> the [linear] "location" value of V
K-> V mod 30^3
Considering a two factor product, take mod 30 of each factor (ie. 01,29) and is labelled as a type.
There exist exactly 8 groups consisting of 4 or 6 "types" as noted within the "equations" textfile.
Exclusive of 2,3 and 5 each group references one of the following primes: 1,7,11,13,17,19,23 and 29.
As an observation note that with the exception of 3, the digit sum of every prime is: 1,4,2,8,5 and 7..lucky 7.


RSA1->RSA8 in the equation sheet references the eight challenge numbers of the defunct [as of 2007] RSA Challenge,eg. RSA7=RSA-1536. From each set of equations,
one of the equations either has factored or will factor the corresponding RSA number.

@Nick- no it's not the best but it works where it should.