When we want to factor in prime numbers the total amount of the sudoku4x4 solutions we obtain

288 = (2 ^ 5) * (3 ^ 2)

But when we want to do the same for sudoku9x9 we get

6,670,903,752,021,072,936,960 = (2 ^ 20) * (3 ^ 8) * 5 * 7 * 27,704,267,971

Which can also be seen as

6,670,903,752,021,072,936,960 = Relabeling * [Invariant Grids]

6,670,903,752,021,072,936,960 = 9! * [18,383,222,420,692,992]

And then obtaining the prime factors of 18,383,222,420,692,992 we obtain

18,383,222,420,692,992 = (2 ^ 13) × (3 ^ 4) × 27,704,267,971

Being able then to write the solutions as

6,670,903,752,021,072,936,960 = 9! * [(2 ^ 13) × (3 ^ 4) × 27,704,267,971]

The obvious question is what internal Sudoku mechanism creates or generates this very large prime number within the Invariant Grids?

The concept map of any size sudoku that I now have in my head tells me that:

n (is the base of sudoku),

--From n arise or emerge the number of cells (in my book "Las Casillas")

---- From the number of cells comes the number of templates (in my book "Las Grillas")

------ From the Templates the Invariants Grids arise (in my book "Los Modelos")

-------- Two things emerge from the Invariants Grids :

---------- 1.- The number of Essentially Different (in my book "Las Familias"), using for this the Geometric Permutations (in my book "Las Transformaciones Geometricas")

---------- 2.- The amount of Sudoku Solutions, Using for this the numerical transpositions or Relabeling (in my book "Las Transposiciones Númericas")

-------- From the multiplication of the Geometric Transformations by the Relabeling the VPT´s. (in my book "Alcance Máximo")

Following this conseptual map, I have an equation or mathematical formula for almost all the elements, but the Invariants Grids and the number of Solutions continue to elude me from a generalization and it is partly the fault of that huge prime number, whom I feel like a stone. inside my shoe.

Starting from:

6,670,903,752,021,072,936,960 = 9! * [(72 ^ 2) * (2 ^ 7) * 27.704.267.971]

Have I ever tried to generalize forms for all sudokus assuming this .....

https://github.com/MiguelQuinteiro/ImagenesSudoku/blob/master/SudokuSolutions.png

But it got me nowhere.

Someone will have a clearer idea of this matter in the forum, what suduko mechanism may be generating such a large prime number involved in the number of invariant grids, if so please share it with this simple mortal.

Greetings