Prime numbers are not random
Prime numbers are not scattered at random: beyond 3, they all live on the 6k ± 1 grid, which is built by simply adding neighbours. The composites of this grid are born from four families whose every birth is known in advance; a prime is the position no family has adopted.
What mathematics establishes
FactEvery prime greater than 3 has the form 6k − 1 or 6k + 1: a number not of this form is even or a multiple of 3. The fact is known and in textbooks; received theory then describes the distribution of primes as irregular and readily speaks of the "mystery of the primes".
The ARC reading
TheoremThe grid builds itself: the sum of two consecutive neighbours on the grid falls exactly on an axis 6m, and the next axis follows from it (Theorem 1). Nothing is hidden about where to look.
TheoremThe composites of the grid, which the corpus calls delfines (descendants), are exactly the products of primes from 5 upward, and they are born from four formulas: k = 6ab + a + b, 6ab − a − b, 6ab + a − b, 6ab − a + b. Each family is a regular progression; there is no fifth (Theorem 2).
TheoremA prime is therefore an exact subtraction: the position the four families do not generate. Whether a position is prime is decided in a finite number of steps, by running the families up to the balance point of the factor pair.
TheoremFor every prime p from 5 upward, p × p − 1 is a multiple of 24: the gap between two prime squares is always a multiple of 24. And the digit sum of the period of 1/p is always a multiple of 9 (1/7 = 0.142857…, sum 27).
Reading"Prime numbers were never hidden; the city map had been thrown away."
What remains open
The corpus settles the position of primes: where they are, how to decide them, exactly which are twins. The asymptotic count, how many there are up to a given height, is a question of another kind: it is reduced to a single lemma, named and declared unresolved.
Further reading
- Volume 4, Number Theory (English summary)
- Chapter 3: the self-generating grid and the four families (in French)
- Volume 4, full PDF (in French)