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.

Read more about this topic: Collatz Conjecture

Famous quotes containing the words methods and/or proof:

“I believe in women; and in their right to their own best possibilities in every department of life. I believe that the methods of dress practiced among women are a marked hindrance to the realization of these possibilities, and should be scorned or persuaded out of society.”
—Elizabeth Stuart Phelps (1844–1911)

“To cease to admire is a proof of deterioration.”
—Charles Horton Cooley (1864–1929)