Machine Learnability as a Measure of Order in Aperiodic Sequences

Tue, 09 Sep 2025 00:00:00 +0800

TL;DR

Prime number distribution is deterministic in definition but behaves statistically like a random process, and this paper uses an image-based machine learning model to measure comparative regularity in prime fields at different regions of an Ulam spiral.
Models trained on blocks from the region around 500m outperform models trained on blocks below 25m, implying there is more easily learnable order in the former region than the latter.
Precision and recall scores diverge by region: the model leans toward identifying prime patterns directly at lower numbers and toward eliminating composites at higher numbers, suggesting it favours different classification strategies depending on scale.
These findings align with number theory conjectures that local randomness in prime distribution regularises after scaling by log x, with averages such as density and arithmetic-progression equidistribution coming to dominate at higher magnitudes.
The authors suggest machine learning accuracy itself can serve as an experimental instrument for number theory, offering a way to probe order that isn’t derived analytically.
The method is noted as potentially useful for investigating patterns in strong and weak primes relevant to cryptography.

Key Takeaways

Number Theory on Eigenform Articles