Bounds on tree distribution in number theory

Abstract

By recursively applying the prime decomposition to the exponents, every natural number determines a rooted planar tree in a canonical way. In particular, trees with only one edge correspond to prime numbers. In this work we investigate the occurrence and the distribution of patterns of trees associated to the natural numbers. Bounds from above and below are proven for certain natural quantities. It is proved that the distance between two consecutive occurrences of the same configuration of trees is unbounded. For any k, there is at least one configuration of trees arising from k consecutive integers that occurs infinitely many times. Dirichlet theorem about primes in arithmetic progressions is generalized to any planar rooted tree. The appearence of equal nonplanar trees associated to k consecutive integers is also investigated. Finally, constraints implied by the repeated occurrence of a given configuration of planar trees are analyzed.