>>6945

Some additional details, and perhaps some clarification, of how the different p formulas can be used to navigate the grid.

Attached pic shows a range of records in (1,145) for both positive and negative x.

The columns p and px represent "pointers" to other "t" values and are calculated as:

p column uses the formula "p+t".

px column uses the formula "p+1-t" for odd e, and "p+2-t" for even e.

In this case, the "p" value is being set to n. So the p column represents moves of 145 + t, and the px column represents moves of 145 + 1 - t.

The two records in the rectangle on the left are the 2nd and 3rd occurrences for a[t] = c, transformed by switching a and n (we can always do this because xx+e = 2na).

From there, the underlined p, px, and t values, show how we are currently able to jump between records using these p and px "pointers".

Within (1,145), there are 4 sequences or chains of records connected by a and b values.

These "pointer" values currently enable moving between 2 of the 4 sequences.

The ability to navigate to the first sequence would lead to a solution.

>>7031

>Rather than being a pathway to a solution

Found some patterns using the odd/even triangle square formulas to create factor records between the -f and e columns that warranted a deeper look.

Recall that:

1) Odd squares are created by 1+8T(u), even squares by 4T(u) + 4T(u-1)

2) an n0 can be calculated from any estimated (x+n)(x+n), c and d values. n0 = sqrt( XPN + c ) - d;

3) the n0 can be tested for validity based on the nn + 2d(n-1) + f - 1 = XPN formula.

4) The difference between the estimated (x+n)(x+n) and the nn formula, is called the remainder 2d(n-1). Or rm2dnm1 for short.

5) A factor record exists where rm2dnm1 = 0.

Attached pics for c85 and c288 show 2 views of where the factor records can be found. The matrix view shows a side by side of -f and e columns and their respective calculations, while the list view is sorted by x descending. The XPN, n0, rm2dnm1 columns are explained above.

The new columns show the interesting patterns and relationships between -f and e.

"rm2dnm1 diff" is calculated as the difference between "-f rm2dnm1" and "e rm2dnm1".

For c85 as an example, the entry record at a matching XPN is found in the -f column at {-15:1:874:41:833:917} where x+n=42. It's related record at x-1 is {4:1:842:40:802:884}. The difference between their rm2dnm1 values is 2. i.e. the "rm2dnm1 diff" column.

Performing that same calculation on -f and e records x-1 apart will yield the same "rm2dnm1 diff" value all the way down to u=10 and record {4:1:222:20:202:244}.

The "rm2dnm1 diff" groups for c85 are 2, 4, 6, 8, 10, 12. There are several properties evident in these groups:

1) each group increments by 2.

2) the number of records within each group decreases.

3) the parity of the rm2dnm1 within each group is always the same.

4) a factor record will never be found when the rm2dnm1 parity is odd.

5) the parities between each group alternate. (even, odd, even, odd, etc.)

And finally, the "rm2dnm1 diff" value will equal 2a from the factor record x^2 + e = 2na formula. This is indicated in the "diff/2 = a[t]" column.

(1/2)

>>7383

Attached pics are a few examples to illustrate that the patterns mentioned previously hold regardless of number size.

For c6107, the matrix and list views show where the first group for "rm2dnm1 diff" = 2 finishes at u=763.

This group of records represents a jump in terms of n values from 2976 to 1450, all of which can be skipped because they will produce the same result. In this example, the last record in the group {23:1:1164349:1525:1162824:1165876} is found where x+n = u from the starting record. The range of records in this first group can be calculated depending on odd/even e and/or odd/even x+n.

For c9705341159, the matrix and list pics attached show the consistency in results where the prime solution record is found. Notice again the "rm2dnm1 diff" value equals 2a.

Being able to determine the start and end ranges for each of these "rm2dnm1 diff" groups could achieve a/4 performance if we process linearly.

Integrating a recursive process with appropriate jumps might be the next step to a solution.

(2/2)

>>7604

>A more generalized formula, that requires further testing:

Attached pic is a test of the u mod formula to verify the triangle base calculations from valid known factors.

Starting from the third factor record for c1020100, (0,9216,481) = {0:9216:1010:960:50:20402}, each known factor represents a square multiple "a" value in aan(n-1).

The formula for each cell is:

( u - mod( u, p ) ) / p

where p is a valid factor of aa shown in the header row, and u is the triangle base starting position in the "u" column.

Green cells indicate an exact match to a valid u value.

Yellow cells indicate a mismatch by 1.

For the case of yellow cells, the correct formula would be: ( u - mod( u, p ) ) / p - 1, though not yet sure when that -1 rule applies.