>>9427

When a <-n, d <-> x+n.

Example:

(23,36,24) = {23:36:78:47:31:197} = 6107; f=134; (x+n)=83; (d+n)=114

(23,31,24) = {23:31:83:47:36:192} = 6912; f=144; (x+n)=78; (d+n)=114

>>9430

new_f = f + 2(n-a)

>>9429

>>9433

In the previous example for swapping a and n from c6107 to c6912, f increases from 134 to 144.

The new f can be calculated as f + 2(n-a).

Another example:

c34117

from:

(261,27,38) = {261:27:184:75:109:313} = 34117; f=108; (x+n)=102; (d+n)=211

swap a and n to:

(261,109,38) = {261:109:102:75:27:395} = 10665; f=-56; (x+n)=184; (d+n)=211

new_f = f + 2(n-a) = 108 + 2*(27 - 109) = -56

>>9436

not yet.

>>9477

sample data.

>>9482

It would, indeed. Though this was never only about factoring the RSA numbers.

>>9619

(118,1,7) = {118:1:143:12:131:157} = 20567; f=169;

(-13,1,7) = {-13:1:91:13:78:106} = 8268; f=196;

There is a consistent way to calculate a "d" that falls between two records in (-1,n) using values from D and C records.

By way of an example, for c6107 where d=78:

D2 = {23:1:83:11:72:96}

d[D2] - d = 83 - 78 = 5

i^2 - c = 84^2 - 6107 = 949

sqrt( 949 ) = 30, remainder = 49

in (-1,5), d=30 falls in the gap between:

{-1:5:23:11:12:44}

{-1:5:55:19:36:84}

C2 = {23:1:6283:111:6172:6396}

x[C2] - d = 111 - 78 = 33

j^2 - c = 112^2 - 6107 = 6437

sqrt( 6437 ) = 80, remainder = 37

in (-1,33), d=80 falls in the gap between:

{-1:33:71:43:28:180}

{-1:33:129:65:64:260}

To generalize:

sqrt( D2.i^2 - c ) in the gap of ( -1, d[D2] - d )

sqrt( C2.j^2 - c ) in the gap of ( -1, x[C2] - d )

>>9641

let's refer to the record d is between in (-1,n) as "DD", representing a second gap that appears to be relevant.

Couple of examples of using q as a process instead of a single multiplier, assuming the prime factors are already known.

The "q" record is created with an a value equal to the minimum of q * a or b. The b value is the maximum.

c6107, c21059917, and c610737593 show cases where the smallest q prime multiplier results in the smallest n value.

For c25185549107, on the other hand, the largest q results in smallest n.

>>9766

We've narrowed down i ranges using D and C. Can consistently find records in (-1,n) within the gap. (though not sure what to do with these yet).

Logical to use similar techniques to narrow down j.

Do a[D] and a[C] point to repeatable records in (0,n)?

>>9773

lower bound on D.j matches initial starting position of the iterative search.

upper bound is the issue.

Not seeing a way to limit further than C.j.

>>9775

always less than x of na.

for i between i[D] and j[C], we're dealing with a subset of semiprimes.

for that same subset, it appears that j can be limited between j[D] and b[D2].

a majority of those can be further limited to a[D2].

This applies to all solved Rsa numbers (except Rsa129 again).

>>9778

d[D2] is a better upper range for j.

There is a connection between roots of unity and being able to navigate through multiple sequences in (e,n).

First pic attached shows valid records in (1,901) for t between 1 and 1802. The sequence column shows related records that can be navigated through using the current t=[mp + t] and t=[mp + offset - t] formulas. Columns p and px, respectively. In this case, where m*p = n = 901.

Roots of unity can be calculated as (i^2) % c = 1, for i between 1 and c-1.

For c901, roots_of_unity(901) = [1,52,849,900].

Second pic attached shows t values and differences required to navigate. Each of these differences can be expressed in terms of one of the roots of unity (52) and/or the d values (30), as indicated on the far right.

These t differences can be used to solve via gcd:

gcd( 106, 901 ) = 53

gcd( 136, 901 ) = 17

And each root of unity between 1 and c-1, less 1, has either a or b as a factor.

prime_factors( 52 - 1 ) = 3 x 17

prime_factors( 849 - 1 ) = 2^4 x 53