Integer factorization by means of the dot product of two vectors

Has anyone come across a factoring method like this one?

To factor an odd number of the form: S^2+R via the dot product of two
vectors,

Create one vector (A, B, C) such that:
A is odd
B is odd
C is a multiple of 4
2AC-B^2 = -1

Examples:
(1,1,0)
(1,3,4)
(1,5,12)
(3,5,4)
(15,11,4)
(3,7,8)

Create another vector (I, J, K) such that:
I is odd
If R is even then J is even
If R is odd then J is odd
K=(J^2+R)/2I

Examples:
(1,2,1) R = -2
(7,4,1) R = -2
(17,40, 47) R = -2
(3,3,1) R= -3
(37,15, 3) R = -3
(11,7,2) R = -5

Let S = (A,B,C) dot (I, J, K) = AI +BJ +CK
R = 2IK-J^2
Then N = S^2 + R

Factoring Example: (A, B, C) = (21, 29, 20), (I, J, K) = (23, 64, 89),
R=2(23)(89) - 64^2 = -2
S = 21(23) +29(64) + 20(89) = 4119
N=S^2 + R = 4119^2 -2 = 16966159

First expression for factors is:
N = [gcd(N, SC + I + CJ + BI)][gcd(N, SC + I - CJ - BI)]
N = [gcd(16966159, 4119(20) + 23 + 20(64) + 29(23))][gcd(16966159,
4119(20) + 23 - 20(64) - 29(23))]
N = [gcd(16966159, 82403 + 1947)][gcd(16966159, 82403 - 1947)]
N = [gcd(16966159, 84350)][gcd(16966159, 80456)]
N = [1687][10057]

Second expression for factors is:
N = [gcd(N, SA+K + BK + AJ)][gcd(N, SA + K - BK - AJ)]
N = [gcd(16966159, 4119(21) + 89 + 29(89) + 21(64))][gcd(16966159,
4119(21) + 89 - 29(89) - 21(64))]
N = [gcd(16966159, 86588 + 3925)][gcd(16966159, 86588 - 3925)]
N = [gcd(16966159, 90513)][gcd(16966159, 82663)]
N = [10057][1687]

So, *** all *** you need to do is find the dot product, given N = S^2
+R.
There are endless numbers of vector pairs that will work.
That doen't help, unfortunately.

A deterministic method to find pairs exists and is equivalent to
Fermat's method
on (C^2)N and (A^2)N. A previous post mentioned work done by Lehmer on
factorization
of multiples of N and the effects of the ratio of the factors of N on
the multiple to be used.
If anyone could provide a link or reference to this or related items,
it would be greatly appreciated.

(CJ + BI)^2 = -R(C^2) + 2ICS + I^2 with R, S known, try pairs of C
and I making a square
>>From first expression for factors:
(20(64)+29(23))^2 = 2(20^2) + 2(23)(20)(4119) + 23^2
1947^2 = 800 + 3789480 + 529 = 3790809

(BK + AJ)^2 = -R(A^2) + 2AKS + K^2 with R, S known, try pairs of A
and K making a square
>>From second expression for factors:
(29(89) + 21(64))^2 = 2(21^2) + 2(21)(89)(4119) + 89^2
3925^2 = 882 + 15396822 + 7921 = 15405625

If you want to experiment with these, I recommend Excel with cell
formulas and
conditional color coding to identify the squares. The patterns are
interesting.

Be sure to use both of these! Certain ratios of the factors of N will
cause very large values
of C and I, or A and K, with the other two being small.
Received on Mon May 1 01:53:52 2006