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Determine whether repeated deterministic updates revisit a state, and optionally find the preperiod and cycle length. Recording all visited states uses memory proportional to the visited orbit. Floyd's cycle-finding algorithm detects repetition with constant extra memory.

An integer factorization algorithm that iterates a simple map such as $x\mapsto x^2+c\mathrm{mod}n$ and searches for a nontrivial greatest common divisor of $n$ and the difference of two iterates. A collision modulo an unknown prime factor can reveal that factor before a collision modulo $n$. A cycle detection schedule such as Floyd's cycle-finding algorithm limits the number of stored iterates. Unsuccessful choices of seed or map can be retried.