General Mathematics

    

Cycles Universels et Cycles Dérivés en 3x +1 Généralisé

Authors: Pierre Lamothe

In Collatz-Kakutani sequences that are generalized to px + q, the beginning x and the end y of a sequence are connected by a diophantine equation pm x - 2d y + qc = 0, where m and d are the numbers of multiplication and division. There is a cycle (x = y) if δ (= 2d - pm) divide qc. It is shown that all c are included in parametric rotation cycles (c1 c2 ... cm) for px + δ, and that the rare numerical cycles (x1 x2 ... xm) derive from them when xi = qci / δ are integers. The universal cycles are purely algebrical but the derived cycles result from a numerical coincidence. Assuming that the possible values of qc mod δ are equiprobable, a formula is given for the ocurrence probability of a derived cycle.

Comments: 10 Pages. En français, 4 tableaux

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Submission history

[v1] 2020-02-28 22:17:24

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