The Collatz Problem and Its Generalizations: Experimental Data. Table 1. Primitive Cycles of $(3n+d)-$mappings.

Abstract : "We study here experimentally, and display the detailed tables of the cyclic structures of dynamical system D_d generated by iterations of the functions T_d acting, for all d ã 1 relatively prime to 6, on positive integers : T_d(n) = n \over 2, if n is even; otherwise T_d(n) =(3n+d) \over 2 In the case d = 1, the properties of the system D =D_1 are the subject of the well-known Collatz, or 3n+1, conjecture. According to Jeff Lagarias, 1990, a cycle of the system D_d is called primitive if its members have no common divisor >1. For every one of 6667 systems D_d, 1?d?19999, we calculate its complete, as we argue, list of primitive cycles. Our calculations confirm, in particular, two long-standing conjectures of Lagarias, 1990, and suggest the plausibility of, and fully confirm several new deep conjectures, Belaga, Mignotte, 2000. Another new conjecture suggested by these calculations and concerning a sharp effective upper bound to the minimal member, or perigee, of a primitive cycle, has been surmised and later proved by the first author, Belaga 2003."
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Pré-publication, Document de travail
2006
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  • HAL Id : hal-00129727, version 1

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Edward G. Belaga, Maurice Mignotte. The Collatz Problem and Its Generalizations: Experimental Data. Table 1. Primitive Cycles of $(3n+d)-$mappings.. 2006. 〈hal-00129727〉

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