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Are the Collatz and abc conjectures related?

Olivier Rozier

TL;DR

This work investigates potential connections between the Collatz and $abc$ conjectures by introducing a Lower Bound Hypothesis (LBH) for Collatz-type sequences and showing that, under the $abc$ conjecture, sharper bounds hold for sequences with mostly odd terms. A key innovation is the notion of $\mu$-hits, a subset of $abc$-hits defined via the $\mu$ function that accounts for the size of prime exponents, and their relation to Collatz dynamics. The paper demonstrates that, generically, the LBH bound improves when $abc$ holds, unless a rare $\mu$-hit is encountered, and establishes a structured link between Collatz iterations and $\mu$-hits. It also uncovers a surprising connection to Wieferich primes: certain $\mu$-hits arise from combining Wieferich primes, suggesting a path to generating large $\mu$-hits and hinting at a broader, probabilistic framework governing both conjectures. Overall, the results point to a deeper, albeit nontrivial, relationship between these central number-theoretic problems and motivate further research into the $\mu$-hit mechanism and its implications.

Abstract

The Collatz and $abc$ conjectures, both well known and thoroughly studied, appear to be largely unrelated at first sight. We show that assuming the $abc$ conjecture true is helpful to improve the lower bound of integers initiating a particular type of Collatz sequences, namely finite sequences of a given length where all terms but one are odd with the usual ``shortcut'' form. To obtain sharper bounds in this context, we are led to consider a small subset of the $abc$-hits. Then, it turns out that Collatz iterations as well as Wieferich primes may be used to find large triples in this subset.

Are the Collatz and abc conjectures related?

TL;DR

This work investigates potential connections between the Collatz and conjectures by introducing a Lower Bound Hypothesis (LBH) for Collatz-type sequences and showing that, under the conjecture, sharper bounds hold for sequences with mostly odd terms. A key innovation is the notion of -hits, a subset of -hits defined via the function that accounts for the size of prime exponents, and their relation to Collatz dynamics. The paper demonstrates that, generically, the LBH bound improves when holds, unless a rare -hit is encountered, and establishes a structured link between Collatz iterations and -hits. It also uncovers a surprising connection to Wieferich primes: certain -hits arise from combining Wieferich primes, suggesting a path to generating large -hits and hinting at a broader, probabilistic framework governing both conjectures. Overall, the results point to a deeper, albeit nontrivial, relationship between these central number-theoretic problems and motivate further research into the -hit mechanism and its implications.

Abstract

The Collatz and conjectures, both well known and thoroughly studied, appear to be largely unrelated at first sight. We show that assuming the conjecture true is helpful to improve the lower bound of integers initiating a particular type of Collatz sequences, namely finite sequences of a given length where all terms but one are odd with the usual ``shortcut'' form. To obtain sharper bounds in this context, we are led to consider a small subset of the -hits. Then, it turns out that Collatz iterations as well as Wieferich primes may be used to find large triples in this subset.
Paper Structure (6 sections, 5 theorems, 47 equations, 2 figures, 2 tables)

This paper contains 6 sections, 5 theorems, 47 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A particular type of Collatz sequences
  3. A rare type of $abc$-hits
  4. From Collatz sequences to $\mu$-hits
  5. Further insights from Wieferich primes
  6. Conclusion

Key Result

Theorem 2.1

Assume that the $abc$ conjecture is true. Then, for every $\varepsilon > 0$, there exists a constant $K(\varepsilon) > 0$ such that

Figures (2)

  • Figure 1: Log plot of the number of misses of Hypothesis \ref{['hyp:LBH']} (LBH) with respect to the constant $C$ arbitrarily fixed between 0 and 1, in the particular case of Collatz sequences of length $j$ having exactly one even term ($q=j-1$) for $j$ up to 1 000, 3 000, 10 000 and 30 000.
  • Figure 2: Log-log plot of the number $N$ of $\mu$-hits (blue), $abc$-hits (red) and good $abc$ triples (green) below $x$. The dashed line corresponds to the power law $N = \left(x / x_0 \right)^{\alpha}$ with $\alpha = 2/11$ and $x_0=1\,000$.

Theorems & Definitions (12)

  • Conjecture 1.1
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['th:LBH_ABC']}
  • Theorem 4.1
  • proof
  • Lemma 5.1
  • proof
  • Theorem 5.2
  • ...and 2 more