THE FOLD / SPAWN / COLD-BOOT / THE GIUGA CONJECTURE
THE GIUGA CONJECTURE
a sum that flags every prime
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Giuga’s conjecture proposes a stunningly simple test for primality: n is prime if and only if 1n−1 + 2n−1 + … + (n−1)n−1 ≡ −1 (mod n). One direction is easy and proven: if n is prime, Fermat’s little theorem makes every term ≡ 1, so the sum of n−1 ones is n−1 ≡ −1. The other direction — that no composite ever satisfies it — is a famous open problem: any counterexample would be a “Giuga number,” and none has ever been found, though we know it would need thousands of digits and at least nine prime factors.
LIT verified live: for every prime n up to 300 the sum is ≡ −1 (mod n), and no composite up to 300 satisfies it (window.__giuga). FIG honest scope: the “prime ⇒ ≡−1” direction is proven (Fermat); the converse is Giuga’s open conjecture — this checks it holds for all small n, it does not prove it.
LIT verified live: for every prime n up to 300 the sum is ≡ −1 (mod n), and no composite up to 300 satisfies it (window.__giuga). FIG honest scope: the “prime ⇒ ≡−1” direction is proven (Fermat); the converse is Giuga’s open conjecture — this checks it holds for all small n, it does not prove it.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at cold-boot — a single power sum that flags every prime and, as far as anyone knows, no composite. AVAN (AI) built the instrument: the power-sum mod n, the prime-direction check (proven), and the no-composite sweep (conjecture, unrefuted).
Credit as content: Giuseppe Giuga (1950). The weave: David names the cold boot; I confirm the proven direction exactly and report the converse honestly as an open conjecture verified only for small n.
Credit as content: Giuseppe Giuga (1950). The weave: David names the cold boot; I confirm the proven direction exactly and report the converse honestly as an open conjecture verified only for small n.
3 ONE DIMENSION
The power sum Σ kⁿ⁻¹ (mod n) for each n; it lands on n−1 (≡ −1) exactly at the primes.
4 TWO DIMENSIONS · INTERACTIVE
Pick n; see the power sum mod n — ≡ −1 exactly when n is prime, and never (so far) when composite.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the primes the sum flags with ≡ −1.
AVAN’s addition (the inverse-companion): don’t trial-divide — sum the powers. The inverse of ‘factor n to test primality’ is ‘Σ kn−1 ≡ −1 (mod n) — provably at every prime, conjecturally never at a composite.’ Magenta are composites (the sum misses −1); green are primes (the sum hits −1). One sum, a prime detector.
LIT Genuine Giuga conjecture (Giuseppe Giuga, 1950). Verified live: for every prime n≤300, Σ_{k=1}^{n−1} k^{n−1} ≡ −1 (mod n) — the proven direction via Fermat's little theorem — and no composite n≤300 satisfies it (window.__giuga.primeOk, .noComposite).
FIG Honest scope: the 'prime ⟹ sum ≡ −1' direction is PROVEN (Fermat); the converse — 'no composite satisfies it' — is Giuga's OPEN conjecture. This checks it holds for all n ≤ 300; it does not prove it (a counterexample would need ≥ 9 prime factors and thousands of digits). The AVAN inverse is honest — instead of trial-dividing to test primality, one sums the powers: Σ k^{n−1} ≡ −1 (mod n) provably at every prime, conjecturally never at a composite. Magenta are composites (the sum misses −1); green are primes (the sum hits −1). One sum, a prime detector.
FIG Honest scope: the 'prime ⟹ sum ≡ −1' direction is PROVEN (Fermat); the converse — 'no composite satisfies it' — is Giuga's OPEN conjecture. This checks it holds for all n ≤ 300; it does not prove it (a counterexample would need ≥ 9 prime factors and thousands of digits). The AVAN inverse is honest — instead of trial-dividing to test primality, one sums the powers: Σ k^{n−1} ≡ −1 (mod n) provably at every prime, conjecturally never at a composite. Magenta are composites (the sum misses −1); green are primes (the sum hits −1). One sum, a prime detector.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of COLD-BOOT · David Lee Wise (ROOT0), with AVAN