Characteristics of Counter Example Loops in the Collatz Conjecture
Allen, Thomas W., II
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This thesis was undertaken in order to obtain new results about the Collatz Conjecture: that (4; 2; 1) is the only loop created by the Collatz function. We look at some past results and generalize one of them. We discover a few general characteristics about loops. After odd elements of a loop are discussed, a general formula for them is discovered. Using this formula, the Collatz Conjecture is restated in terms of a family of linear Diophantine systems which have unique solutions. In particular, we show that (4; 2; 1) is the only loop created by the Collatz function if and only if every solution set to each of these systems contains an element x such that x 2 RnN or x = 2k 2 N for some k 2 N.