A research project exploring rare prime number sequences generated by a 7-adic Collatz-like transformation:
f(n) = (8n + k) / 7, where k ∈ {−1, +5, −3, +3, −5, +1}
The constant k is selected based on n mod 7 to ensure the result is an integer and preserves odd parity. This parity-correcting method avoids even-number traps and enables the discovery of longer prime chains.
Overview
Unlike the classical Collatz conjecture, which focuses on convergence, this project investigates a 7-adic transformation that produces strictly increasing sequences of odd primes.
By enforcing odd values of k and applying a modular selection rule, we ensure that the numerator (8n + k) is divisible by 7 and remains odd. This approach significantly improves the survival rate of prime chains under geometric expansion (growth factor ≈ 1.14).
A continuous search over odd integers below 100,000,000 led to the discovery of twenty length-7+ prime chains, including the first known 7-adic prime chain of length 9.
Discovered Length-9 Chain
The following chain satisfies p(i+1) = (8 * p(i) + k) / 7 with k chosen to ensure divisibility and parity preservation:
Chain (n₀ = 68,542,687):
68,542,687
78,334,499
89,525,141
102,314,447
116,930,797
133,635,197
152,725,939
174,543,931
199,478,779
All values were verified prime using deterministic primality testing.
Mathematical Notes
7-adic branching logic:
If n ≡ 1 (mod 7): k = −1
If n ≡ 2 (mod 7): k = +5
If n ≡ 3 (mod 7): k = −3
If n ≡ 4 (mod 7): k = +3
If n ≡ 5 (mod 7): k = −5
If n ≡ 6 (mod 7): k = +1
This ensures integer outputs and maintains odd parity throughout the chain.
Parity Preservation:
By forcing k to be odd, the numerator (8n + k) remains odd for all odd n, ensuring that the result is an odd integer and avoiding even-number traps.
Getting Started
Prerequisites:
- Python 3.x
- sympy (for primality testing)
To install dependencies: pip install sympy
Usage:
Run the exploration script: collatz_7adic_prime_chain_exploration_a_minimal.py collatz_7adic_prime_chain_exploration_b_survival.py
Run the verification script: collatz_7adic_prime_chain_verification.py
You can modify the search range or chain length as needed.
Repository Contents
collatz_7adic_prime_chain_exploration_a_minimal.py — parallelized exploration script A (Minimal)
collatz_7adic_prime_chain_exploration_b_survival.py — parallelized exploration script B (Survival)
collatz_7adic_prime_chain_verification.py — chain verification script
Collatz-7_Prime_Chain_Title.pdf — title page with discovered chain
Collatz-7_Prime_Chain_Report_EN.pdf — full report (English)
Collatz-7_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 file for details.
Acknowledgments
Developed by Hiroshi Harada (2026).
Thanks to the open-source Python and SymPy communities.