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 of, methods and/or proof:

“Methods of thought which claim to give the lead to our world in the name of revolution have become, in reality, ideologies of consent and not of rebellion.”
—Albert Camus (1913–1960)

“The greatest part of our faults are more excusable than the methods that are commonly taken to conceal them.”
—François, Duc De La Rochefoucauld (1613–1680)

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