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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) ?
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    I have tried to tex it, but changed some notation ($r_0$, instead of $R_0$). Hope that does not confuse the comments above.2013-04-03

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