that computes in base two Repeated applications of the Collatz function can be represented as an abstract machine that handles strings of bits. The machine will perform the following three steps on any odd number until only one remains: Append to the (right) end of the number in binary (giving ); Add this to the original number by binary addition (giving ); Remove all trailing s (that is, repeatedly divide by 2 until the result is odd). Example The starting number 7 is written in base two as . The resulting Collatz sequence is: 111 1111 10110 10111 100010 100011 110100 11011 101000 1011 10000 As a parity sequence For this section, consider the Collatz function in the slightly modified form This can be done because when is odd, is always even. If is the parity of a number, that is and , then we can define the Collatz parity sequence (or parity vector) for a number as , where , and . Which operation is performed, or , depends on the parity. The parity sequence is the same as the sequence of operations. Using this form for , it can be shown that the parity sequences for two numbers and will agree in the first terms if and only if and are equivalent modulo . This implies that every number is uniquely identified by its parity sequence, and moreover that if there are multiple Hailstone cycles, then their corresponding parity cycles must be different. Applying the function times to the number will give the result , where is the result of applying the function times to , and is how many increases were encountered during that sequence. For example, for there are 3 increases as 1 iterates to 2, 1, 2, 1, and finally to 2 so the result is ; for there is only 1 increase as 1 rises to 2 and falls to 1 so the result is . When is then there will be rises and the result will be . The factor of 3 multiplying is independent of the value of ; it depends only on the behavior of . This allows one to predict that certain forms of numbers will always lead to a smaller number after a certain number of iterations: for example, becomes after two applications of and becomes after 4 applications of . Whether those smaller numbers continue to 1, however,