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

Edward G. Belaga (1) , Maurice Mignotte (1)

1 IRMA - Institut de Recherche Mathématique Avancée

"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."

Mots clés: "Collatz problem; discrete dynamical system; cyclic and divergent trajectories; insolubility"; Collatz problem; insolubility

Domaines: Théorie des nombres [math.NT]

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