A conceptual exploration by Satoshi Sakamori
This repository presents a conceptual analogy between the distribution of prime numbers and the structure of irrational numbers.
Both sets can be interpreted as residual structures that remain when all regular, periodic, or rationally describable subsets are systematically removed from the number line.
- Primes: remain after eliminating all periodic sets of multiples (via the sieve of Eratosthenes).
- Irrationals: remain after removing all rational fractions (via rational partitions of the real line).
This idea was first described informally by Satoshi Sakamori and then expressed in LaTeX form as an initial working paper.
The core note is written in LaTeX:
The PDF is a concise conceptual note intended to invite mathematical and philosophical discussion rather than present a formal theorem.
| Domain | Regular structure removed | Residual set | Interpretation |
|---|---|---|---|
|
|
Multiples of smaller numbers (periodic sets) | Prime numbers | Multiplicative aperiodicity |
|
|
Rational fractions |
Irrational numbers | Additive/rational aperiodicity |
Both can be seen as non-periodic residues of rational or modular orderings.
- Define an entropy-like measure of “irregularity” for the prime indicator function.
- Compare with the entropy of continued fraction coefficients of typical irrationals.
- Explore whether a shared measure of rational deficiency exists between
$\mathbb{N}$ and$\mathbb{R}$ .
This is an open discussion draft.
Feedback, critique, and extensions are very welcome.
- 💬 Open an Issue for conceptual comments or related ideas.
- 🔀 Send a Pull Request if you’d like to add mathematical formalisms, visualizations, or references.
Please keep the discussion constructive and exploratory — this is a speculative mathematical idea, not a formal claim.
This repository is released under the Creative Commons Attribution 4.0 (CC BY 4.0) license.
You are free to share and adapt the material with attribution.
Satoshi Sakamori (坂森聡)
Toyama, Japan
X (Twitter): @satoshisakamori
“Both primes and irrationals mark the limits of order —
what remains when the world’s periodicity runs out.”