Collatz Conjecture - Methods of Proof

Methods of Proof

There have been many methods of attack on the problem. For example, let A and B be integers, A being how many times the "3n+1" rule is used in a cycle, and B being how many times the "n/2" rule is used. Let x be the lowest number in a cycle then, regardless of what order the rules are used, we have:

$\frac{3^A}{2^B}x + C = x$

where C is the "excess" caused by the "+1" in the rule, and can be shown to be bigger than:

$C \ge \frac{3^{A-1}}{2^B}$

using geometric progression. Rearranging shows that the lowest number in the cycle satisfies:

$x \ge \frac{3^{A-1}}{2^B-3^A}$

which gives a lower bound for the lowest number in a cycle for a given cycle length. For large cycles the fraction 3A/2B would be expected to tend to 1, so that the lower bound would be large.

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