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:
where C is the "excess" caused by the "+1" in the rule, and can be shown to be bigger than:
using geometric progression. Rearranging shows that the lowest number in the cycle satisfies:
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:
“A woman might claim to retain some of the child’s faculties, although very limited and defused, simply because she has not been encouraged to learn methods of thought and develop a disciplined mind. As long as education remains largely induction ignorance will retain these advantages over learning and it is time that women impudently put them to work.”
—Germaine Greer (b. 1939)
“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)
“A short letter to a distant friend is, in my opinion, an insult like that of a slight bow or cursory salutation—a proof of unwillingness to do much, even where there is a necessity of doing something.”
—Samuel Johnson (1709–1784)