Possible generalization (example):

23=(7)3+(2);

7=(2)3+(1);

2=(0)3+(2);

0=(0)3+(0);

Take only remainders, and form a vector a=(2 1 2 0) What can be said about components of the vector for the prime next to p? E.g., do i-th components repel, like the 1-st ones?