primes4everybody (p4e) implements a fully transparent and reproducible model of the prime distribution:
The number space is not a static infinite object.
It grows additively — and is reconstructed multiplicatively.
Prime numbers are the positions where multiplicative reconstruction cannot keep pace with additive growth.
This repository allows anyone to reproduce, verify or falsify the theory — without needing advanced mathematics.
| Classical mathematics | p4e perspective |
|---|---|
| Numbers exist all at once | Numbers appear step by step |
| Multiplication is neutral | Multiplication imposes interference patterns |
| Primes are anomalies | Primes are structural growth-gaps |
| π(n) looks mysterious | π(n) follows from deterministic growth energetics |
The core mechanism:
Additive growth is complete and gap-free.
Multiplicative reconstruction is incomplete and generates interference.
The gaps of this interference correspond exactly to prime numbers.
This removes the conceptual mystery around primes without using sieving, factorization, modular arithmetic, or probability.
Primes appear exactly at the positions where multiplicative coverage fails to occupy the next additive step.
Unlike classical approaches, primality is not determined by performing any computation on x.
Whether x is prime is already decided by the global multiplicative state before x is reached.
Each row represents a prime emitter (multiplicative coverage).
Each column represents the additive growth of the number space.
Black gaps between coverage waves correspond exactly to the primes.
Growth density increases.
Multiplicative coverage increases.
Remaining gaps become narrower — but never disappear.
This explains why π(n) slows down:
not because primes "run out", but because reconstruction pressure rises.
The repository contains two implementations, serving different purposes.
This version is intentionally minimal.
It implements the constructive mechanism in the simplest possible form:
- additive growth
- multiplicative reconstruction using local emitters
- emergence of primes when coverage fails
Its purpose is:
- teaching
- visualization
- transparency
- falsifiability
- step-by-step understanding
It is not optimized for performance — by design.
The MCG is a full structural generator of the theory:
- unbounded prime generation
- segmented number space
- deterministic global multiplicative coverage state
- no sieving, no modulus, no trial division
- high performance despite zero divisibility checks
- faithful expression of the mathematical architecture
-
Unbounded:
Generates primes indefinitely without restarting. -
Segmented growth:
Processes the number space in fixed-size segments
(default: 1,000,000 integers per segment). -
Generative dynamics:
Each new segment is determined by the multiplicative waves
of all previously discovered primes. -
Fast:
On a standard machine, the MCG produces
10,000 primes in under one second
— without performing a single divisibility test. -
Scientifically reproducible:
The algorithm is a direct operationalization of the theory
from the accompanying research paper.
go run Multiplicative-Coverage-Generator.goThis prints the first 10,000 primes using the unbounded constructive generator.
(See README for tree)
The model is incorrect if any of the following occur:
| Failure | Meaning |
|---|---|
| A composite is marked prime | Model is wrong |
| A prime is missed | Model is wrong |
| Growth diverges significantly from π(n) | Model is incomplete |
| No Riemann-oscillation behaviour appears | Model is incomplete |
The project is valuable only because it can be disproven.
The Irreducible Structure of the Prime Distribution
Zenodo: https://zenodo.org/records/17649211
License: CC BY-NC-ND 4.0
This is not:
- a new sieve
- a primality test
- a cryptographic trick
- a probabilistic guess
It is a constructive explanation of why primes exist and
how they are distributed — not as anomalies, but as natural consequences of growth.
Primes for everybody.
MIT License — maximum openness for research and education.