I am trying to express my extended multiplication pixel-by-pixel, and to do that I need to extend the information held by a pixel. Define a periodic pixel to be a 2-list of pixels [[a,b],[c,d]]. This periodic pixel represents the regular pixels [a,b], [a+c,b+d], [a+2c,b+2d], ..., but I don't know if [a-c,b-d] should be included here. Let's restrict ourselves at first to periodic pixels for which [1,1]<=[a,b]<=[c,d] so that we don't need to worry about that. Then [[a,b],[c,d]] is the same as the equation det([[i-a,j-b],[c,d]]) = 0, where we look for pixels [i,j].

EDIT: The determinant equation is not enough since that would make det([[i-a,j-b],[cq,dq]]) = q det([[i-a,j-b],[c,d]]) but I want [[i-a,j-b],[cq,dq]] to have less solutions than [[i-a,j-b],[c,d]]. So the rest of this comment is WRONG.


Defining [[a,b],[c,d]] [[e,f],[g,h]] will be a matter of reducing a pair of equations, namely
>>    det([[i-a,j-b],[c,d]]) = 0

>>    det([[j-e,k-f],[g,h]]) = 0

to a single equation of the same form,
>>    det([[i-x,k-y],[z,w]]) = 0.
This is a congruential problem. I have partially solved it. This is my most recent equation:
>>    ids+bM+fgr = kgr+eM+ads
where [r,s] = [c,h]/gcd(c,h) and [m, M] = [gcd, lcf](c, h) = [gcd(c, h), ch/gcd(ch)]. If all that notation is too much for you, it's basically
>>    idh+bch+fgc = kgc+ech+adh.


How do we find x,y,z,w from here?


Sometimes there is no solution. A classical example is [[a,b],[c,d]] = [[e,f],[g,h]] = [[1,2],[2,2]], to which we associate the matrix [0,1;0,0]. There is no solution because the square of this matrix is the zero matrix. A novel example is [[a,b],[c,d]] = [[e,f],[g,h]] = [[1,4],[2,6]] (basically [0,0,0,1,0,0;0,0,0,0,0,0]) which has to do with the definition of the collatz sequence 4, 1, 10, 2, 16, 3, ... .


Suggestions?