Let $N, K, W$ be natural numbers
If I start from $R_0$, say any integer $r_0, 0 \lt r_0 \lt N$
and proceed with:
$R_j = ( R_{j-1} + K ) \mod W,\quad j=1,2, \dots$
(that is the remainder of the division $\frac{R_{j-1} + K }{W}$).
I can imagine that, when $R_j$ "returns" to the initial value $r_0$, a new "cycle" of identical remainders will start again: $R_0 = r_0 , R_1 , \dots , R_{p-1} = r_0$
Can we say something more about this sequence of remainders (does it have a name, etc.) ? In particular:
- What is the (minimal) length $p$ of the cycle (period) in general ?
- Can we express $R_j$ in a "non recursive" form ( $R_j$ = some function of the initial data) ?