The Sufficiency of Arithmetic Progressions for the 3x+1 Conjecture

Define T : N+ - N+ by T(x) = (3x + 1) /2 if x is odd and T (x) = x/2 if x is even. The 3x + 1 Conjecture states that the T-orbit of every positive integer contains 1. A set of positive integers is said to be sufficient if the T-orbit of every positive integer intersects the T-orbit of an element of that set. Thus to prove the 3x + 1 Conjecture it suffices to prove it on some sufficient set. Andaloro proved that the sets 1 + 2n N are sufficient for n ≤ 4 and asked if 1 + 2n N is also sufficient for larger values of n. We answer this question in the affirmative by proving the stronger result that A + BN is sufficient for any nonnegative integers A and B with B ≠ 0, i.e. every nonconstant arithmetic sequence forms a sufficient set. We then prove analagous results for the Divergent Orbits Conjecture and Nontrivial Cycles Conjecture