This project explores prime generation patterns from twin primes by applying linear offsets to the first element (p) of each twin prime pair (p, p+2).
An earlier version of this analysis tested 4 offsets. Expanding to 7 offsets produces a significantly higher prime yield of 82.03%, accounting for 7.65% of total primes analyzed.
For each twin prime pair (p, p+2), the following expressions were tested for primality:
- q1 = 2p + 1
- q2 = 2p + 3
- q3 = 2p - 3
- q4 = 2p - 5
- q5 = 2p + 7
- q6 = 2p + 9
- q7 = 2p - 9
- Twin primes tested: all pairs with (p ≤ 1,000,000).
- Prime testing: each (q) was checked for primality using Python.
- Analysis goal: measure how often each offset produces primes and compare across offsets.
- Expanding from 4 to 7 offsets substantially increased the number of generated primes.
- Certain offsets consistently outperform others in prime yield.
- This suggests nontrivial distribution patterns worth further study.
(A full results table is available in the repository outputs.)
- All Python scripts used for generating and testing primes are included.
- Data files contain counts and distributions of primes by offset.
If you use this work, please cite via Zenodo:
Open questions and possible directions:
- Why do certain offsets consistently yield m