THE FOLD / CHEAT / THE ROOT KIT / THE FERMAT PRIMES
THE FERMAT PRIMES
five in a row, then Euler
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Fermat looked at 3, 5, 17, 257, 65537 — the numbers 2^(2ⁿ)+1 — found every one of them prime, and wrote in 1640 that he was convinced they all were, while admitting he could not prove it. Ninety-two years later Euler took the sixth one apart. F₅ = 4,294,967,297 = 641 × 6,700,417, and he found it not by trial division but by narrowing the search: any factor of Fₙ must be congruent to 1 modulo 2^(n+2), which cut the candidates for F₅ to a short list. In the four centuries since, not one further Fermat prime has ever been found — the tally is still exactly five, and the modern suspicion runs the opposite way: that no others exist.
LIT verified live in exact BigInt: F₀…F₄ are all prime by deterministic Miller–Rabin; 641 × 6,700,417 = F₅ exactly and F₅ fails primality; Euler’s sieve rule checks out — both factors of F₅ are 1 mod 128 = 2^(5+2); and Landry’s 1880 factorisation 274,177 × 67,280,421,310,721 = F₆ multiplies out exactly (window.__fermatprimes).
LIT verified live in exact BigInt: F₀…F₄ are all prime by deterministic Miller–Rabin; 641 × 6,700,417 = F₅ exactly and F₅ fails primality; Euler’s sieve rule checks out — both factors of F₅ are 1 mod 128 = 2^(5+2); and Landry’s 1880 factorisation 274,177 × 67,280,421,310,721 = F₆ multiplies out exactly (window.__fermatprimes).
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-root-kit — the cheat: Euler did not break in by brute force, he read the specification and found the constraint every factor had to satisfy. Knowing the shape of the key beats guessing keys. AVAN (AI) built the instrument: the BigInt Fermat-number ladder, deterministic primality, and the exact factorisation checks.
Credit as content: Pierre de Fermat (1640, the conjecture); Leonhard Euler (1732, F₅; 1747, the 2^(n+2) rule); Fortuné Landry (1880, F₆). This sphere is the first in WORLD II to carry a DEAD stamp — a scheme taken from David’s own rev5 instrument, which grades claims LIT (measured), AMBER (assigned), DEAD (tested, disproven). The weave: David names the root kit and supplies the stamp; I run the arithmetic that killed the conjecture.
Credit as content: Pierre de Fermat (1640, the conjecture); Leonhard Euler (1732, F₅; 1747, the 2^(n+2) rule); Fortuné Landry (1880, F₆). This sphere is the first in WORLD II to carry a DEAD stamp — a scheme taken from David’s own rev5 instrument, which grades claims LIT (measured), AMBER (assigned), DEAD (tested, disproven). The weave: David names the root kit and supplies the stamp; I run the arithmetic that killed the conjecture.
3 ONE DIMENSION
The ladder: five primes, then the wall at F₅.
4 TWO DIMENSIONS · INTERACTIVE
Walk Euler's sieve: only 1-mod-128 candidates survive.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the doubling tower, five lamps lit and the rest dark.
AVAN’s addition (the inverse-companion): don’t count the confirmations — ask how many cases you could even reach. The inverse of ‘five in a row’ is ‘five was the whole feasible sample’: Fermat checked every case his arithmetic could hold and generalised from a sample of five. Magenta is F₅, the first case he could not compute; green is the five he could. A pattern that spans your entire budget is not evidence about what lies past it.
LIT Verified live in exact BigInt: F₀…F₄ all prime by deterministic Miller–Rabin; 641 × 6,700,417 = F₅ exactly and F₅ fails primality; Euler's sieve rule holds — both factors are 1 mod 128; and Landry's 274,177 × 67,280,421,310,721 = F₆ multiplies out exactly (window.__fermatprimes.ok).
FIG Fermat, Euler and Landry credited as content. This is the first sphere in WORLD II to carry a DEAD stamp — the scheme comes from David's own rev5 instrument, which grades claims LIT (measured) / AMBER (assigned) / DEAD (tested, disproven). The AVAN inverse — ask how many cases you could even REACH: Fermat generalised from a sample of five because five was his entire arithmetic budget. A pattern spanning your whole budget says nothing past it.
DEAD Fermat's conjecture that every Fn is prime. Killed by Euler in 1732 at the very first case Fermat could not compute. Exactly five Fermat primes are known and the modern expectation is that there are no more.
FIG Fermat, Euler and Landry credited as content. This is the first sphere in WORLD II to carry a DEAD stamp — the scheme comes from David's own rev5 instrument, which grades claims LIT (measured) / AMBER (assigned) / DEAD (tested, disproven). The AVAN inverse — ask how many cases you could even REACH: Fermat generalised from a sample of five because five was his entire arithmetic budget. A pattern spanning your whole budget says nothing past it.
DEAD Fermat's conjecture that every Fn is prime. Killed by Euler in 1732 at the very first case Fermat could not compute. Exactly five Fermat primes are known and the modern expectation is that there are no more.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE ROOT KIT · David Lee Wise (ROOT0), with AVAN