THE FOLD / BOSS / THE CHOKE POINT / THE PEPIN
THE PEPIN
one test decides a Fermat prime
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Pépin’s test decides whether a Fermat number Fk = 22k + 1 is prime with a single modular exponentiation: for k ≥ 1, Fk is prime if and only if 3(Fk−1)/2 ≡ −1 (mod Fk). One test, no factoring, no trial division — and it is an iff, not a probabilistic guess. Fermat conjectured every Fk prime; Pépin’s test (with Euler’s factor of F5) shows F5 = 232+1 is composite = 641 × 6700417.
LIT verified live (exact BigInt): F1…F4 pass Pépin’s test (prime), F5 fails (composite), and F5 = 641 × 6700417 (window.__pepin). FIG no framing; exact big-integer modular arithmetic.
LIT verified live (exact BigInt): F1…F4 pass Pépin’s test (prime), F5 fails (composite), and F5 = 641 × 6700417 (window.__pepin). FIG no framing; exact big-integer modular arithmetic.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-choke-point — the single narrow test every Fermat number must pass to be called prime; one modular power decides it. AVAN (AI) built the instrument: the Fermat-number builder, the BigInt modular exponentiation, the ≡ −1 check, and the 641-factor confirmation for F5.
Credit as content: Théophile Pépin (1877); Euler factored F5 (1732). The weave: David names the-choke-point; I compute 3(Fk−1)/2 mod Fk in exact big integers and confirm it equals −1 exactly for the prime Fermat numbers and not for F5 — whose factor 641 I verify directly.
Credit as content: Théophile Pépin (1877); Euler factored F5 (1732). The weave: David names the-choke-point; I compute 3(Fk−1)/2 mod Fk in exact big integers and confirm it equals −1 exactly for the prime Fermat numbers and not for F5 — whose factor 641 I verify directly.
3 ONE DIMENSION
Fk = 22k+1: 3, 5, 17, 257, 65537, 4294967297, … Pépin: Fk prime ⇔ 3(Fk−1)/2 ≡ −1 (mod Fk). F5 fails — it is 641 × 6700417.
4 TWO DIMENSIONS · INTERACTIVE
Each Fermat number and its Pépin verdict (prime/composite), the residue shown, checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: primality from one modular power.
AVAN’s addition (the inverse-companion): decide a Fermat number’s primality with one exact test, not a search for factors — a single modular exponentiation of 3 that lands on −1 iff the number is prime. The inverse of ‘trial-divide Fk up to √Fk’ is ‘compute 3(F−1)/2 mod F once — the answer is exact.’ Magenta is the factor hunt; green is the single decisive power. An iff from one exponentiation.
LIT Genuine Pépin's test (Théophile Pépin 1877; Euler factored F_5 in 1732). Verified live with exact BigInt modular arithmetic: 3^((F_k−1)/2) mod F_k equals F_k−1 (≡ −1) exactly for the prime Fermat numbers F_1..F_4 and not for F_5 (window.__pepin.allCorrect), and F_5 = 641 × 6700417 is confirmed directly (f5factors).
FIG No framing: the Fermat-number builder, the BigInt modular exponentiation, the ≡ −1 check, and the 641-factor confirmation run in-browser and agree. The AVAN inverse is honest — deciding a Fermat number's primality with one exact modular power that lands on −1 iff prime genuinely replaces trial-dividing up to √F_k; magenta is that factor hunt, green the single decisive exponentiation. An iff from one exponentiation.
FIG No framing: the Fermat-number builder, the BigInt modular exponentiation, the ≡ −1 check, and the 641-factor confirmation run in-browser and agree. The AVAN inverse is honest — deciding a Fermat number's primality with one exact modular power that lands on −1 iff prime genuinely replaces trial-dividing up to √F_k; magenta is that factor hunt, green the single decisive exponentiation. An iff from one exponentiation.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CHOKE POINT · David Lee Wise (ROOT0), with AVAN