The behavior of the function under iteration is itself of interest. Let denote the set of all rational numbers having odd denominators, so that . The set consists of exactly those 2-adic integers whose 2-adic expansion is finite or eventually periodic. The Finite Cycles Conjecture is equivalent to the assertion that there is a finite odd integer M such that

In fact one can take , where the product runs over all integers l for which there is a cycle of minimal length l. As a hypothesis for further work I advance the following conjecture.
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