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:
“If you want to know the taste of a pear, you must change the pear by eating it yourself.... If you want to know the theory and methods of revolution, you must take part in revolution. All genuine knowledge originates in direct experience.”
—Mao Zedong (1893–1976)
“It would be some advantage to live a primitive and frontier life, though in the midst of an outward civilization, if only to learn what are the gross necessaries of life and what methods have been taken to obtain them.”
—Henry David Thoreau (1817–1862)
“There are some persons in this world, who, unable to give better proof of being wise, take a strange delight in showing what they think they have sagaciously read in mankind by uncharitable suspicions of them.”
—Herman Melville (1819–1891)