Abstract
A "3x+1" cycle of length k occurs when the Collatz function T(n), which takes odd integers n to (3n+1)/2 and even integers n to n/2, applied to an initial integer a0, reach that initial value again after k iterations, so that T^(k)(a0)= a0. It is conjectured that any cycle must have k=⌈i log_2(3)⌉ where i is the number of odd elements in the cycle. It is easy to show that in cycles where a0 is the smallest integer, i<3a0 implies k=⌈i log_2(3)⌉. This paper will show that in cycles, i<304a0 implies k=⌈i log_2(3)⌉. In m-cycles m<1.8296017a0 implies k=⌈i log_2(3)⌉.
