THE FOLD / SPAWN / GENESIS BLOCK / THE SIEVE OF ATKIN
THE SIEVE OF ATKIN
primes from quadratic forms, not multiples
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The sieve of Atkin finds primes using quadratic forms instead of marking multiples. A number (with the small primes handled separately) is prime if it is a solution to one of three modular equations an odd number of times: 4x²+y² (for n mod 12 ∈ {1,5}), 3x²+y² (n mod 12 = 7), or 3x²−y² with x>y (n mod 12 = 11) — then the multiples of prime squares are removed. It is asymptotically faster than Eratosthenes.
LIT verified live: the sieve of Atkin’s prime list up to 5000 is identical to trial division (669 primes) — window.__atkin. FIG no framing; exact.
LIT verified live: the sieve of Atkin’s prime list up to 5000 is identical to trial division (669 primes) — window.__atkin. FIG no framing; exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at genesis-block — the generation of the primes from scratch, here by counting solutions to quadratic forms rather than crossing off multiples. The sieve of Atkin is that generation. AVAN (AI) built the instrument: the three quadratic-form toggles by residue mod 12, the square-multiple removal, and the equals-trial-division check.
Credit as content: A. O. L. Atkin & Daniel Bernstein (2004). The weave: David names genesis-block; I toggle candidates by the parity of their quadratic-form solution counts, strip prime-square multiples, and confirm the primes match trial division exactly.
Credit as content: A. O. L. Atkin & Daniel Bernstein (2004). The weave: David names genesis-block; I toggle candidates by the parity of their quadratic-form solution counts, strip prime-square multiples, and confirm the primes match trial division exactly.
3 ONE DIMENSION
Toggle n whenever a quadratic form (4x²+y², 3x²+y², or 3x²−y²) equals it, subject to n’s residue mod 12. An odd number of hits marks a candidate; then remove multiples of prime squares.
4 TWO DIMENSIONS · INTERACTIVE
The sieve of Atkin’s primes up to a limit; the set is checked against trial division.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: primes found by quadratic forms.
AVAN’s addition (the inverse-companion): find primes by counting solutions to quadratic forms (mod 12) rather than crossing off multiples — an odd solution-count flags a candidate, then prime-square multiples are removed. The inverse of ‘mark every multiple of every prime (Eratosthenes)’ is ‘toggle by quadratic-form parity, then strip squares.’ Magenta is the multiple-marking of Eratosthenes; green is the quadratic-form sieve. Primes from parabolas, not multiples.
LIT Genuine sieve of Atkin (Atkin & Bernstein 2004). Verified live: toggling candidates by the parity of their quadratic-form solution counts (mod 12) and removing multiples of prime squares produces exactly the same primes up to 5000 as trial division (669 primes) — window.__atkin.matchesTrial.
FIG No framing: the three quadratic-form toggles by residue mod 12, the square-multiple removal, and the equals-trial-division check run in-browser and match exactly. The AVAN inverse is honest — counting solutions to quadratic forms (odd count flags a candidate) then stripping prime-square multiples finds primes without marking every multiple; magenta is the multiple-marking of Eratosthenes, green the quadratic-form sieve. Primes from parabolas, not multiples.
FIG No framing: the three quadratic-form toggles by residue mod 12, the square-multiple removal, and the equals-trial-division check run in-browser and match exactly. The AVAN inverse is honest — counting solutions to quadratic forms (odd count flags a candidate) then stripping prime-square multiples finds primes without marking every multiple; magenta is the multiple-marking of Eratosthenes, green the quadratic-form sieve. Primes from parabolas, not multiples.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GENESIS BLOCK · David Lee Wise (ROOT0), with AVAN