Questions -Factoring

From: Aubrey Hutchison (abhjrpe_at_comcast.net)
Date: 09/28/03 

Date: Sun, 28 Sep 2003 09:25:40 -0400

Given:
Primes p and q, unknown to us, has product N which we know.
N is say 8 digits or larger.
Value ratio of p and q are unknown
p and q are assumed to be greater than 1
p is smaller than q

Known:
If LSD of N is a 1,3,7,or 9 the number N is most likely is a product of two
prime numbers or a prime number it self.
The equation of the relationship is N=pq thus
The following 5 points on the curve are known (1,N),(N,1), (p,q),(q,p), and
(sqrt(N),sqrt(N)). Lets consider the curve a floating point curve.
The above points will fall exactly on the curve of the equation as a
Floating point curve.
:
Likewise using integer math division for example, the results will also fall
exactly on the curve even when considered a floating point curve.

Now. if we use integer math points say p+/- 1 will not fall exactly on the
floating point curve, but will generate a curve that is close to the
floating point curve and will match it at only the 5 points listed above.

Next if we look at the rate of changes of the two curves the math landscape
becomes very interesting over the range of 1 to sqrt(N).

The landscape seems to be divided into regions with similar patterns.
The regions are:
sqrt(N) down to sqrt(N/2)
sqrt(N/2) down to sqrt(N/3)
sqrt(N/3) down to sqrt(N/4).......with each region becoming smaller as we
approach 1.
The rate of change of the landscape appears as ripples that appear
saw-toothed in shape like a slope with a cliff at the peak. Starting
downward toward 1
the ripples are spaced widely and as we approach a region boundary they
bunch up. On a number of attempts it appears a Factor is at the bottom of a
cliff.

The pattern of the ripples are not understand but seemto be related to the
value of the factors.

A routine to study the landscape yielded one that finds the harmonic of
factors. If looking at 8 times the value of a factor the routine will
indicate the
actual value of a factor at 1/8th the observed value.

If we can find a factor by finding a harmonic can we develops a fast
factoring routine?