The autoconjugacy of the 3x+1 function Original Research Article

The 3x+1 map T is defined on the 2-adic integers by T(x)=x/2 for even x and T(x)=(3x+1)/2 for odd x and the 3x+1 conjecture states that the T-orbit of any positive integer contains 1. We define and study properties of the unique nontrivial autoconjugacy Ω of T. This autoconjugacy sends x to the unique 2-adic integer whose parity vector is the oneʹs complement of the parity vector of x. We prove that if Ω maps rational numbers to rational numbers then there are no divergent T-orbits of positive integers. The map Ω is then used to restate the 3x+1 conjecture in a parity neutral form. We derive a necessary and sufficient condition for a cycle to be self-conjugate and show that self-conjugate cycles contain only positive elements. It is then shown that the only self-conjugate cycle of integers is {1,2}. Finally, we prove that for any rational 2-adic integer x, lim̄κn(x)n+limκn(Ω(x))/n=1 where κn(x) is the number of ones in the first n digits of the parity vector of x, and we use this along with generalizations of known restrictions on lim̄ κn(x)/n to prove most of the results in the paper.