Pics attached for a random example c19367 and it's na transform to c321535 where F takes up the entire (x+n)(x+n).
>hundreds and hundreds of bitmaps. And tables.
Still working on analyzing these bitmaps and tables and trying to find patterns that will enable larger n movements than n+4.
Depending on f mod 8, the middle (n-1)+(f-1) "square" can be assembled using either 1 or 2 different sized triangles. And sometimes a remainder that is different between the triangles by 1.
The assumption here, is that if we can determine the formulas to construct these middle squares, those formulas can be used to scale into the full x+n square we are looking for.
The latest thoughts in narrowing down these rules revolves around analyzing mods for the various components of nn + (2d-1)(n-1) + (n-1) + (f-1) formula.
Attached examples for x+n=29 show the mod analysis and animated squares grouped by f mod 8 values of 1 and 5.
>if we can determine the formulas to construct these middle squares, those formulas can be used to scale into the full x+n square we are looking for
Attached pics further the f mod 8 analysis mentioned in the previous posts.
For each f mod 8 value of 1, 2, 5, and 6, these tests group a starting c record with it's solution prime record to enable a comparison of the mod values for nn, (2d-1)(n-1), (n-1), and (f-1).
Based on these tables, a consolidate mod breakdown has been created that may show the various possible triangle combinations needed for a more accurate iterative search. (fmod_analysis_wip4.png)
For example, if the c starting record can be described as f mod 8 = 1 and (n-1) mod 8 = 0, it's possible that the solution prime record can be found with one of two triangle combinations:
4T(u1) + 4T(u2), or 8T(u1)
Still to be determined if the (n-1)+(f-1) mod accurately describes the triangle ratios, and if so, the correct u1 and u2 values to use.
Initial work exploring the up and down movements.
Pics attached are two factor examples for c159, c291, c295, and c303 and represent both the (1,c) starting record and (a,b) prime solutions.
For these examples (moving up/down), e, f, d and c are the same. The n and x+n squares are different.