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Collatz conjecture becomes theorem

Grażyna Mirkowska, Andrzej Salwicki

Abstract

The Collatz hypothesis is a theorem of the algorithmic theory of natural numbers. We prove the (algorithmic) formula that expresses the halting property of Collatz algorithm. The observation that Collatz's theorem cannot be proved in any elementary number theory completes the main result.

Collatz conjecture becomes theorem

Abstract

The Collatz hypothesis is a theorem of the algorithmic theory of natural numbers. We prove the (algorithmic) formula that expresses the halting property of Collatz algorithm. The observation that Collatz's theorem cannot be proved in any elementary number theory completes the main result.
Paper Structure (23 sections, 33 theorems, 56 equations, 11 figures, 2 tables)

This paper contains 23 sections, 33 theorems, 56 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. An illustration of some regularities.
  3. Properties of the structure $\mathfrak{N}$ of the natural numbers.
  4. A theory that completely specifies the structure $\mathfrak{N}$.
  5. Properties of computations
  6. Collatz tree
  7. Halting conditions for Collatz's computations
  8. Hotel Collatz $\mathcal{HC}$
  9. The tree $\mathcal{F}$ of triples.
  10. The graph $\mathcal{G}$ of odd, natural numers
  11. The diagram of the $3n+1$ structure of odd numbers
  12. Main lemma
  13. The limitations of the Main lemma
  14. The positive consequnces of main lemma.
  15. Proof of the Collatz conjecture
  16. ...and 8 more sections

Key Result

Lemma 3.1

The following algorithm Gr is equivalent to Collatz algorithm $Cl$.

Figures (11)

  • Figure 1: An example of compact computation
  • Figure 2: A fragment of Collatz tree
  • Figure 3: Strata $W_{0} -- W_{4}$ of Collatz tree
  • Figure 4: Tree of triples (strata 4 -- 15)
  • Figure 5: Hotel Collatz
  • ...and 6 more figures

Theorems & Definitions (59)

  • Definition 1.1
  • Definition 1.2
  • Definition 2.1
  • Conjecture 1
  • Lemma 3.1
  • Corollary 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 49 more