After reflection, it is in fact the case that if there is a loop that begins with a number other than 1, call it z, that the sequence of operations that cause z to loop around to z will in fact converge on z regardless of the number chosen as a starting point. The number z is encoded in the sequence of operations and if that sequence is repeated any starting number will converge on Z.
The combination of operations that make up the Collatz conjecture, i.e multiplying odd numbers by three and adding 1 or dividing even numbers by zero, can be described by a linear function of the form
y = ax + b.
More accurately, for the Collatz conjecture, y = (ax + b)/c where a is equal to 3^N, where N is the number of odd numbers that appear in the loop, and c = s^ m, and represents the total number of divisions by 2 performed in traversing the loop. The term b is a number that is determined solely by the sequence of operations and is only implicitly dependent on z. The equation above describes a line with slope a/c and y intercept of b/c. This line will intersect the line y = x at the point (z,z); thus for any sequence that can be written in the form above, any starting number n will converge on the value b/(c-a). Also, all sequences, regardless of whether they terminate in a loop containing 1 or some other number, will converge on a number b/(c-a). For example, the Collatz sequence that begins with number 7 terminates at 1, can be described by the equation (243x + 347)/2048. If you begin with any number n and repeat the sequence many times, the result will converge on 347/(2048-243) = 0.192243. Likewise, if you take the sequence describing the loop based on 1; (3x+1)/4, begin with any number, not necessarily an integer and substitute each result back into the equation, the results will converge on 1.
This result is so because any number less than z will increase when substituted into an equation that loops, and any number greater than z will decrease.. This can be shown graphically.
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