THE FOLD / CHEAT / THE ROOT KIT / THE LUCKY EULER
THE LUCKY EULER
a polynomial that spits primes forty times in a row
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Euler’s lucky numbers come from a startling coincidence he found in 1772: the polynomial n² + n + 41 produces a prime for every n from 0 to 39 — forty primes in an unbroken run: 41, 43, 47, 53, 61, 71, …, 1601. The streak finally breaks at n = 40, where 40²+40+41 = 1681 = 41². Even beyond that it stays astonishingly prime-rich (about 58% of values up to n = 1000 are prime). The magic isn’t luck: 41 is the largest of the six ‘lucky numbers of Euler’, tied to the fact that the imaginary quadratic field of discriminant -163 = 1 - 4·41 has class number one — unique factorization, the deepest reason the primes line up.
LIT verified live: n²+n+41 is confirmed prime for all n = 0 to 39, composite at n = 40 (equal to 41²), and about 58% of values up to n = 1000 are prime (window.__luckyeuler). FIG no framing; the polynomial values and their primality are computed independently in-browser.
LIT verified live: n²+n+41 is confirmed prime for all n = 0 to 39, composite at n = 40 (equal to 41²), and about 58% of values up to n = 1000 are prime (window.__luckyeuler). FIG no framing; the polynomial values and their primality are computed independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-root-kit — the cheat: a single quadratic that injects forty primes in a row, no sieve required. AVAN (AI) built the instrument: the polynomial n²+n+41, its forty-prime streak, and the break at 41².
Credit as content: Leonhard Euler (1772); the connection to discriminant -163 and class number one. The weave: David names the prime-cheat; I confirm the forty-long streak and its exact break.
Credit as content: Leonhard Euler (1772); the connection to discriminant -163 and class number one. The weave: David names the prime-cheat; I confirm the forty-long streak and its exact break.
3 ONE DIMENSION
n²+n+41 for n = 0…44 — an unbroken run of green primes, breaking to red at n = 40 (=41²).
4 TWO DIMENSIONS · INTERACTIVE
Step n; n²+n+41 is factored and tested — prime through n=39, then 41² at n=40.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the forty-long streak of primes from one quadratic.
AVAN’s addition (the inverse-companion): don’t marvel at the streak — ask why it holds. The inverse of ‘forty primes in a row’ is ‘discriminant -163 has class number one — unique factorization forces it’. Magenta are the polynomial values; green is the unbroken run of primes they form. A coincidence that is really a deep theorem.
LIT Genuine Euler's lucky numbers / prime-generating polynomial (Leonhard Euler, 1772; discriminant −163, class number one). Verified live: n²+n+41 is prime for all n = 0 to 39, composite at n = 40 (= 41²), and about 58% of values up to n = 1000 are prime (window.__luckyeuler.allPrime, .fail40, .cnt).
FIG No framing; the polynomial values and their primality are computed independently in-browser. The AVAN inverse is honest — instead of marvelling at the streak, ask why it holds: the inverse of 'forty primes in a row' is 'discriminant −163 has class number one — unique factorization forces it'. Magenta are the polynomial values; green is the unbroken run of primes they form. A coincidence that is really a deep theorem.
FIG No framing; the polynomial values and their primality are computed independently in-browser. The AVAN inverse is honest — instead of marvelling at the streak, ask why it holds: the inverse of 'forty primes in a row' is 'discriminant −163 has class number one — unique factorization forces it'. Magenta are the polynomial values; green is the unbroken run of primes they form. A coincidence that is really a deep theorem.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE ROOT KIT · David Lee Wise (ROOT0), with AVAN