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Bounds on tree distribution in number theory

Roberto Conti, Pierluigi Contucci, Vitalii Iudelevich

Abstract

The occurrence and the distribution of patterns of trees associated to natural numbers are investigated. Bounds from above and below are proven for certain natural quantities.

Bounds on tree distribution in number theory

Abstract

The occurrence and the distribution of patterns of trees associated to natural numbers are investigated. Bounds from above and below are proven for certain natural quantities.
Paper Structure (6 sections, 22 theorems, 195 equations, 1 figure)

This paper contains 6 sections, 22 theorems, 195 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Some bounds for $\kappa_+(n)$
  3. The distance between same trees
  4. Trees in arithmetic progressions
  5. Same tree on consecutive numbers
  6. When two configurations of trees are equal

Key Result

Theorem 1

Let $x$ be large enough and $1\leqslant k \leqslant \log x/(5\log\log x)$. Then there are $m\geqslant x^{1/10}$ numbers $n_1, n_2, \ldots, n_m\leqslant x$ such that $\kappa_+(n_1), \kappa_+(n_2), \ldots, \kappa_+(n_m)\geqslant k.$

Figures (1)

  • Figure 1: Labeled tree representation of the integer $320=2^{6}\cdot 5=2^{2\cdot 3}\cdot 5$

Theorems & Definitions (43)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 33 more