Computational Investigations of the Collatz Conjecture

Verification · Cycle Bounds · Stopping Times · Trajectory Families

Ayazelrico

Research Presentation

Computational Investigationsof the Collatz Conjecture

Verification to \(2^{36}\), rigorous lower bounds on nontrivial cycles, negative integer cycles, and the 2-adic perspective — by Ayazelrico.

90-Day Research Program — Active

A structured daily research program is underway. Every claim is labelled Kanıtlandı / Hesaplandı / Bilinen literatür / Varsayım·Sezgi.

Day 01 published: Site Audit & Epistemic Label System →

Abstract

The Collatz conjecture asserts that iterated application of the map \(T(n)=n/2\) if \(n\) is even and \(T(n)=(3n+1)/2\) if \(n\) is odd eventually reaches 1 for every positive integer \(n\). This work presents computational methods and rigorous lower bounds by Ayazelrico.

Every integer in \(2\le n<2^{36}\) has been verified to descend below its starting value (Hesaplandı). Continued-fraction analysis yields a lower bound of at least 190 537 odd elements (total length \(\ge 492\,531\)) for any nontrivial cycle (Kanıtlandı). Three negative cycles are recovered; the 2-adic perspective is discussed.

1. Introduction

\[T(n)=\begin{cases}n/2 & n\equiv 0\pmod{2},\\(3n+1)/2 & n\equiv 1\pmod{2}.\end{cases}\]

The geometric-mean heuristic factor is approximately \(\sqrt{3}/2\approx 0.866<1\) (Varsayım/Sezgi) — still only a heuristic, not a proof.

2. Principal Results

Verification to \(2^{36}\) · Hesaplandı

Every integer \(n\) with \(2\le n<2^{36}\) eventually falls below its starting value. Run time: 157 seconds.

Lower Bound on Nontrivial Cycles · Kanıtlandı

Any nontrivial cycle with elements \(\ge 2^{36}\) must contain at least 190 537 odd terms (total length \(\ge 492\,531\)).

Verified bound \(B\)Min. odd \(k\)Min. total lengthSource
\(2^{30}\)47 468\(\ge 122\,703\)Literatür
\(2^{36}\) (this work)190 537\(\ge 492\,531\)Bu proje
\(2^{68}\)\(8.96\times 10^{9}\)\(\ge 2.3\times 10^{10}\)Literatür (Barina et al.)
\(2^{71}\)\(7.17\times 10^{10}\)\(\ge 1.85\times 10^{11}\)Literatür (Barina et al.)

3. Overview of Methods

Inductive Verification

Block-table acceleration. Verification for all \(n<2^{36}\).

Details →

Cycle Lower Bounds

Continued fractions of \(\log_2 3\) yield rigorous length bounds.

Details →

Stopping-Time Statistics

First-descent histograms and tail density.

Details →

Families & Negatives

Union-find clustering; three negative cycles; 2-adic view.

Details →

4. 90-Day Program

A daily research program is active. Each day produces a new page under strict epistemic labelling.

5. Scope and Limitations

Verification holds only up to \(2^{36}\) (Hesaplandı). The cycle bound does not prove absence of cycles or divergent trajectories. A genuine proof must exploit positivity (Varsayım/Sezgi).

6. Author

Research by Ayazelrico. Official site: collatz-computational-research.ayazelrico.com.

Selected References

  1. Lagarias, J. C. (ed.). The Ultimate Challenge: The 3x+1 Problem. AMS, 2010.
  2. Terras, R. “A stopping time problem on the positive integers.” Acta Arith. 30 (1976).
  3. Tao, T. “Almost all orbits of the Collatz map attain almost bounded values.” Forum Math. Pi (2019).
  4. Barina, D. and subsequent verification projects.
  5. Oliveira e Silva, T. Empirical verification records.