>>6972

Just posting a shortcut using the "pointer" formulas to help identify the various record sequences in any (e,n).

Attached pics for (1,61), (1,145), (-24,60), and (0,36) show a range of records in both positive and negative x values.

A new "p root" column has been added, which is calculated simply as t mod n.

The resulting value can be used to navigate to the first record in a sequence or as a way to identify the number if unique sequences.

For example, in (0,36), there are 6 sequences shown with "p root" values of 1,7,13,19,25,36.

>>7383

>>7384

Pics attached are summary views for c145, c629, c6107 and c15120, to further illustrate the groupings for rm2dnm1 between -f and e columns.

Each example is grouped by the "rm2dnm1 diff" value, and then filtered to only show even parities. Just looking for any obvious patterns for larger jumps.

Have also confirmed that if the same calculation is performed on u values above the c entry record, then the rm2dnm1 value will always equal c.

>>7411

May have stumbled across a way to find some factor records for trivial cases using only e and d (or at the very least confirmed an avenue worth exploring).

Pics attached are annotated matrix views showing side by side -f and e columns for c85, c145, c287, and c459. All records are included for c85 and c145, while only matching factor records are shown for the others.

In (e,1), we know that f = (x+n)(x+n), or sqrt(f) = x+n. And because n=1, x can always be expressed in terms of f as x = sqrt(f) - 1.

These small number test cases seem to indicate that there is a relationship between records where a record lower in the column can be found at d = sqrt(f).

For the c145 example where the starting x+n = 72, a corresponding record in (e,1) with the same x+n is found at {1:1:2592:71:2521:2665}. At this record, f = 5184 and sqrt(f) = 72.

Lower down in (e,1), we find the record {1:1:72:11:61:85} where d = 72, the same value as the previous record's sqrt(f). The resulting (x+n)(x+n) from this record is then used to solve the problem.

For c85, however, this jumping by d = sqrt(f) requires an extra iteration.

Starting at {-15:1:874:41:833:917}, f=1764, we calculate the lower d value as sqrt(1764) = 42 and jump down to {-15:1:42:9:33:53}, f=100. If a solution isn't found, we repeat the calculation for the next lower d as sqrt(100) = 10, and jump down to {-15:1:10:5:5:17}, f=36.

For c287 - similar to c145 - a solution can be found in a single calculation using d = sqrt(f) where f is the (x+n)(x+n) from the starting 1,c record.

And for c459, this method finds 3 out of the 4 factor records. d value jumps from 26229 -229 -> 21.