The Set of Cycles For The 3x + 1 and 3x + k Problems

This paper studies the set of cycles under iteration of the (3x + k)-function T sub k (n) = sup 3n+k over 2 if n is odd, n over 2 if n is even, where k varies over all positive integers relatively prime to 6. This set of cycles is large and has an interesting structure. A cycle of T sub k (.) is primitive if all its members are relatively prime to k. There is a natural bijection between the variation of all such cycles and the set variations of rational cycles of the (3x + 1)-function T sub 1 (.), acting on the domain of all rational numbers with odd denominator. Let C sub (prim) (k) count the number of primitive cycles of T sub k (.) and C sub (prim) (k,y) the number of primitive cycles of period =y. It is conjectured that C sub (prim) (k) >= 1 and C sub (prim) (k) inf for all k. Both these conjectures seem difficult to settle. This paper studies the more tractable function C sub (prim (k,y). It shows that C sub (prim) (k,k sup (1/3)) = 0 infinitely often, while numerical evidence suggests that C sub (prim) (k,k sup (1+epsilon) >= 1 for all large k.