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:

“The reading public is intellectually adolescent at best, and it is obvious that what is called “significant literature” will only be sold to this public by exactly the same methods as are used to sell it toothpaste, cathartics and automobiles.”
—Raymond Chandler (1888–1959)

“If any doubt has arisen as to me, my country [Virginia] will have my political creed in the form of a “Declaration &c.” which I was lately directed to draw. This will give decisive proof that my own sentiment concurred with the vote they instructed us to give.”
—Thomas Jefferson (1743–1826)