THE FOLD / LOOT / THE VAULT / THE WIEFERICH
THE WIEFERICH
the vanishingly rare Wieferich primes
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Wieferich primes are primes p so rare that only two are known. Fermat’s little theorem says 2p−1 ≡ 1 (mod p) for every odd prime; a Wieferich prime satisfies the far stronger congruence modulo p²: 2p−1 ≡ 1 (mod p²). Only 1093 and 3511 qualify below 6.7×1015 — despite vast searches, no third is known. They are tied to Fermat’s Last Theorem: any prime exponent counterexample of the first case would have to be Wieferich.
LIT verified live (exact BigInt): among all primes below 20000, exactly 1093 and 3511 satisfy 2p−1 ≡ 1 (mod p²) (window.__wieferich). FIG no framing; exact big-integer modular exponentiation.
LIT verified live (exact BigInt): among all primes below 20000, exactly 1093 and 3511 satisfy 2p−1 ≡ 1 (mod p²) (window.__wieferich). FIG no framing; exact big-integer modular exponentiation.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-vault — the deepest lock, holding the vanishingly rare primes that pass Fermat’s test one level stronger, modulo p². Wieferich primes are that vault. AVAN (AI) built the instrument: the BigInt 2p−1 mod p² test and the exhaustive scan finding exactly 1093 and 3511.
Credit as content: Arthur Wieferich (1909). The weave: David names the-vault; I raise 2 to the p−1 modulo p² in exact big integers and confirm that among all primes under 20000, only 1093 and 3511 land on 1 — the rarest of primes.
Credit as content: Arthur Wieferich (1909). The weave: David names the-vault; I raise 2 to the p−1 modulo p² in exact big integers and confirm that among all primes under 20000, only 1093 and 3511 land on 1 — the rarest of primes.
3 ONE DIMENSION
Every odd prime: 2p−1 ≡ 1 (mod p). Wieferich: 2p−1 ≡ 1 (mod p²) — a much rarer coincidence. Only 1093 and 3511 are known.
4 TWO DIMENSIONS · INTERACTIVE
A prime and its Fermat quotient mod p²; the two Wieferich primes stand out, checked over a range.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: primes passing Fermat one level deeper.
AVAN’s addition (the inverse-companion): sharpen Fermat’s test from mod p to mod p² — asking not just that 2p−1 leave remainder 1 modulo p, but modulo p² — and almost no prime survives. The inverse of ‘2p−1 ≡ 1 (mod p) always’ is ‘does it hold mod p² — the vanishingly rare Wieferich condition?’ Magenta is Fermat mod p (every prime); green is Fermat mod p² (only two known). Rarity from one more power.
LIT Genuine Wieferich primes (Arthur Wieferich 1909). Verified live with exact BigInt modular exponentiation: among all primes p below 20000, exactly 1093 and 3511 satisfy 2^(p−1) ≡ 1 (mod p²) (window.__wieferich.onlyKnown) — the only two Wieferich primes known anywhere (none found below 6.7×10¹⁵).
FIG No framing: the BigInt 2^(p−1) mod p² test and the exhaustive scan finding exactly 1093 and 3511 run in-browser and agree. Honest scope: this confirms the two known Wieferich primes below 20000 — it does not (and cannot) settle whether infinitely many exist, an open problem. The AVAN inverse is honest — sharpening Fermat's test from mod p (every prime) to mod p² (almost none) is the genuine Wieferich condition; magenta is Fermat mod p, green Fermat mod p². Rarity from one more power.
FIG No framing: the BigInt 2^(p−1) mod p² test and the exhaustive scan finding exactly 1093 and 3511 run in-browser and agree. Honest scope: this confirms the two known Wieferich primes below 20000 — it does not (and cannot) settle whether infinitely many exist, an open problem. The AVAN inverse is honest — sharpening Fermat's test from mod p (every prime) to mod p² (almost none) is the genuine Wieferich condition; magenta is Fermat mod p, green Fermat mod p². Rarity from one more power.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE VAULT · David Lee Wise (ROOT0), with AVAN