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:
“All men are equally proud. The only difference is that not all take the same methods of showing it.”
—François, Duc De La Rochefoucauld (1613–1680)
“With a generous endowment of motherhood provided by legislation, with all laws against voluntary motherhood and education in its methods repealed, with the feminist ideal of education accepted in home and school, and with all special barriers removed in every field of human activity, there is no reason why woman should not become almost a human thing. It will be time enough then to consider whether she has a soul.”
—Crystal Eastman (1881–1928)
“If some books are deemed most baneful and their sale forbid, how, then, with deadlier facts, not dreams of doting men? Those whom books will hurt will not be proof against events. Events, not books, should be forbid.”
—Herman Melville (1819–1891)