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The sequence xn =2n mod b x n = 2 n mod b becomes eventually cyclic because it has the form xn+1 = 2xn x n + 1 = 2 x n, that is to say xn+1 = f(xn) x n + 1 = f (x n) and the set of remainders modulo b b is finite. However this alone does not solve the problem for the original sequence.
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