The Collatz Problem and Its Generalizations: Experimental Data. Table 2. Factorization of Collatz Numbers $2^l-3^k$.

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

1 IRMA - Institut de Recherche Mathématique Avancée

The purpose of the present paper is to provide the reader with the table of the factorization in primes of all Collatz numbers $2^\ell-3^k>0$,in the interval $1 <\ell<115$. The interest of such experimental data is double. First, Collatz numbers represent a natural and, in a sense, minimal generalization of two classes of integers, Cunningham integers and Schinzel integers, with the experimental factorization of Cunningham integers representing an ongoing, well-known and well-organized project, initiated by John Brillhart, D. H. Lehmer, J. L. Selfridge, Bryant Tuckerman, and Sam S. Wagstaff. Second, Collatz numbers play the crucial role in the Diophantine interpretation of the Collatz problem : one of the most interesting rephrasing of this problem claims that no narrow Collatz number could be a divisor of numbers from a certain finite set of natural numbers called the Collatz corona.

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

Domaines: Théorie des nombres [math.NT]

Edward G. Belaga, Maurice Mignotte. The Collatz Problem and Its Generalizations: Experimental Data. Table 2. Factorization of Collatz Numbers $2^l-3^k$.. 2006. ⟨hal-00129730⟩