Working in binary protects the repetends of 1/3h: Comment on Colussi's ‘The convergence classes of Collatz function’

• Colussi (2011) described the binary strings that iterate to 1 under the Collatz map.
• We confirm Colussi's description via elementary methods.
• Colussi's strings hold the repetends of 1/3h1/3h, optionally rotated and replicated.
• It is only in binary that we see the repetends of 1/3h1/3h.
• We see the Collatz map as multiplying up a fraction, versus reducing a large number.

The Collatz function can be stated as ‘for any odd positive integer x  , calculate 3x+13x+1 and then divide by 2 until the result is odd’. Colussi (2011) discovered and proved that if x attains 1 on the k  th iteration of the Collatz function, then its binary representation can be written as the concatenation of strings sksk−1…s1sksk−1…s1 where each shsh is a finite and contiguous extract from the representation of 13h. We provide an elementary confirmation of Colussi's finding, and comment on how working in binary ‘protects’ the repetends of 13h as formed into each shsh.