Here's some accidental mathematical art in Excel. Each cell is a²b-b mod c where b is the column, c is the row, and a varies from 1 to 30. I put a heat map on the cells (small=red, big=green) and noticed the beautiful patterns.

those moire patterns seem to be fragments of fourier transforms which are (more than merely surgically related), and are also produced in some cases when complex functions are plotted at too low a resolution

Try this: for every cell (x,y), color it blue if and only if its image under 30 degree rotation around the origin is within 0.2 of a grid point (integer coordinates). It’s beautiful and unexpected, like a mathematical Persian carpet.

I plotted the equation on a larger scale with c,b∈[0,1000] and a ∈ [2,30]. Without any proof, it graphically have 3-period fixed points, which is a flag for chaotic behavior. It also demonstrates SDIC properties and seems to have a dense orbit on the natural numbers.

Is there something special about numbers of the form a^2-1 for this? What about just ab mod c?

This feels very Wolfram automata!

Lovely!

there's probably a good gateway to analytic number theory through studying why these patterns exist

how are you getting it to change over time though? please share more details!