A research project exploring rare prime number sequences generated by a 3-adic Collatz-like map:
If n ≡ 1 (mod 3): f(n) = (4n − 1) / 3
If n ≡ 2 (mod 3): f(n) = (4n + 1) / 3
If n ≡ 0 (mod 3): stop
This repository documents the discovery of a length-8 prime chain, found through a full search up to 6×10^8.
While the classical Collatz conjecture concerns eventual convergence to 1,
local behaviors of Collatz-like maps can produce increasing sequences that remain prime for several steps.
In this project, we study a 3-adic linear fractional transformation that always returns an integer and branches depending on n mod 3.
Restricting the orbit to primes yields a generalized Cunningham chain.
A complete computational search revealed a unique 8-term prime chain within the tested range.
The following 8 consecutive primes satisfy
p(i+1) = (4*p(i) ± 1) / 3,
with the sign determined by p(i) mod 3:
46000363
61333817
81778423
109037897
145383863
193845151
258460201
344613601
This is the only maximal chain of length ≥ 8 found below 6×10^8.
The map depends entirely on n mod 3:
- 1 → (4n − 1)/3
- 2 → (4n + 1)/3
This ensures integrality and creates a branching structure absent in classical Cunningham chains.
The map grows approximately as:
f(n) ≈ (4/3)n
Combined with the decreasing density of primes, long chains become extremely rare.
No other chain of length 8 or longer was found in the entire search range.
Many shorter chains merge into this orbit, but none extend beyond it.
- Python 3.x
sympyfor primality testing
Install dependencies:
pip install sympy
Run the exploration script:
python collatz_3adic_prime_chain_exploration.py
Run the verification script (to check a specific chain):
python collatz_3adic_prime_chain_verification.py
You can modify the search range or explore for longer chains.
collatz_3adic_prime_chain_exploration.py— full-range exploration scriptcollatz_3adic_prime_chain_verification.py— chain verification / testing scriptresults/— discovered chains and logsREADME.md— project documentationLICENSE— MIT License
A formal record of this discovery is archived on Zenodo.
This project is licensed under the MIT License.
See the LICENSE file for details.
Developed by Hiroshi Harada (2026).
Thanks to the open-source Python and SymPy communities.