From sokol at livecamserver.com Wed Jan  2 08:50:13 2002
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From: "John Sokol" <sokol at livecamserver.com>
Cc: "Jesse Monroy" <jessem at livecamserver.com>
Subject: I may have figured out a fast way to factor the product of 2 big primes.!
Date: Wed, 2 Jan 2002 08:58:07 -0000

Definitiions:

Primorials
P2: 2 * 3 = 6
P3: 2 * 3 * 5 = 30
P4: 2 * 3 * 5 * 7 = 210
P5: 2 * 3 * 5 * 7 * 11 = 2310

Pn * m is as good as any primorial
and
Pn + Pm is as good as any primorial
Where n and m and any whole number.
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Sokol's conjecture:
A primes can only exists + or - a prime from a primorial.
Where 1 is considered a prime and 2 is not.
Ie. 1,3,5,7,11,13 ...
P3 = (2 * 3 * 5) = 30

Primes     13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 , 47
30 +      -17 ,-13 ,-11 , -7 . -1 , +1 , +7 ,+11 ,+13 ,+17

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% = Modulo function

But I consider it's output to have positive and negative components.
Meaning 8 % 10 = 8 I consider this as also (10 - 8) = +8 and -2
and 12 % 10 = 2  can be considered  -8 and +2. Lets call -8 Aneg and +2
Apos

SOLUTION:

I have 2 big primes X and Y
Z = X * Y

Where Z % X = 0

Z % Pn = +A and -A   I just say  A = Z % Pn

Now factor:
for (n = 2 to ?)
for ( x = 1 to ? )
Factor( Absolute Value(Aneg) + pn * x )
Factor( Apos + Pn * x)

Until Factor = two primes.

These primes are + or - Distances from Pn * x that had  2 primes factors return.
(The x is not the same as the for loop hit on.)

EXAMPLE:

Let X = 19 and Y = 37

Z = 19 * 37 = 703

( primorial 19 - 30 = -11 and  37 - 30 = +7 )

A = Z % Pn * x

703 % 30 = 13 ( or - 17)

30*2 + 17 = 77  : 77 is factors 11 and 7

30 - 11 = 19 and    30 + 7 = 37

Now this is messy and requires systematicaly tring Both + and - for various values on n and x, but this is still reduces the solution space a vast amount.