A research project exploring rare prime number sequences generated by a 6-adic Collatz-like transformation:
f(n) = (7n + k) / 6, where k ∈ {−1, −5, +5, +1}
The constant k is selected based on n mod 12 to ensure the result is an integer and remains odd. This parity-preserving method avoids even-number traps and enables the discovery of long prime chains.
Overview
While the classical Collatz conjecture focuses on convergence, this project investigates a 6-adic transformation that produces strictly increasing sequences of odd primes.
By applying a modular rule to select k ∈ {±1, ±5}, we ensure that the numerator (7n + k) is divisible by 6 and remains even. This approach significantly increases the survival rate of prime chains under geometric expansion.
A continuous search over odd integers below 10,000,000 led to the discovery of nine length-7 prime chains and one length-8 chain.
Discovered Length-8 Chain
The following chain satisfies:
p(i+1) = (7 * p(i) + k) / 6
with k chosen to ensure divisibility and parity preservation.
Chain (n₀ = 1,099,687):
1,099,687 1,282,969 1,496,797 1,746,263 2,037,307 2,376,859 2,773,003 3,235,171
All values were verified prime using deterministic primality testing.
Mathematical Notes
6-adic branching logic:
If n ≡ 1 (mod 12): f(n) = (7n − 1) / 6 If n ≡ 5 (mod 12): f(n) = (7n − 5) / 6 If n ≡ 7 (mod 12): f(n) = (7n + 5) / 6 If n ≡ 11 (mod 12): f(n) = (7n + 1) / 6
This ensures integer outputs and maintains odd parity throughout the chain.
Parity Preservation:
By choosing k ∈ {±1, ±5}, the numerator (7n + k) remains even for all odd n, ensuring the result is an odd integer and avoiding even-number traps.
Getting Started
Prerequisites:
Python 3.6 or higher
sympy (for primality testing)
To install dependencies:
pip install sympy
Usage:
To run the exploration script:
python hunt_6adic.py
To run the verification script:
python verify_6adic_chain.py
You can modify the search range or chain length as needed in the script.
Repository Contents
collatz_6adic_prime_chain_exploration.py — Parallelized exploration script
collatz_6adic_prime_chain_verification.py — Chain verification script
Collatz-6_Prime_Chain_Title.pdf — Title page with discovered chain
Collatz-6_Prime_Chain_Report_EN.pdf — Full report (English)
Collatz-6_Prime_Chain_Report_JP.pdf — Full report (Japanese)
README.txt — Project documentation
LICENSE.txt — MIT License
Citation
A formal record of this discovery is archived on Zenodo.
License
This project is licensed under the MIT License. See the LICENSE.txt file for details.
Acknowledgments
Developed by Hiroshi Harada (2026). Thanks to the open-source Python and SymPy communities.