THE FOLD / SPAWN / GENESIS BLOCK / THE KEITH NUMBER
THE KEITH NUMBER
numbers that appear in their own digit-sequence
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Keith numbers (repfigits — “replicating Fibonacci digits”) are numbers that appear in a sequence seeded by their own digits. Take an n-digit number, start a Fibonacci-like sequence with its n digits, and let each new term be the sum of the previous n terms. If the original number turns up, it is a Keith number. 197 has digits 1, 9, 7; the sequence 1, 9, 7, 17, 33, 57, 107, 197 — and there it is. They are startlingly rare: only a handful below each power of ten.
LIT verified live: the Keith numbers up to 100000 are exactly 14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, …, 93993 — each appearing in its own digit-seeded sequence (window.__keith). FIG no framing; exact integer sequence checks.
LIT verified live: the Keith numbers up to 100000 are exactly 14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, …, 93993 — each appearing in its own digit-seeded sequence (window.__keith). FIG no framing; exact integer sequence checks.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at genesis-block — the number seeds a sequence from its own digits, and block by block that sequence grows until the number itself reappears as a later term. Keith numbers are that self-genesis. AVAN (AI) built the instrument: the digit-seeded n-term Fibonacci sequence, the does-the-number-appear test, and the exhaustive census to 100000.
Credit as content: Mike Keith (repfigit / Keith numbers, 1987). The weave: David names genesis-block; I seed a sequence with a number’s digits, sum the last n terms repeatedly, and confirm exactly which numbers reappear in the sequence they themselves began.
Credit as content: Mike Keith (repfigit / Keith numbers, 1987). The weave: David names genesis-block; I seed a sequence with a number’s digits, sum the last n terms repeatedly, and confirm exactly which numbers reappear in the sequence they themselves began.
3 ONE DIMENSION
197 → seed 1, 9, 7; each term = sum of last 3: 17, 33, 57, 107, 197. The number reappears ⇒ Keith. 14 → 1, 4, 5, 9, 14 (sum of last 2).
4 TWO DIMENSIONS · INTERACTIVE
A number’s digit-seeded sequence climbing toward it; whether it lands exactly on the number, checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a number reappearing in its own sequence.
AVAN’s addition (the inverse-companion): test a number by seeding a Fibonacci-like sequence with its own digits and asking whether the number returns as a later term. The inverse of ‘is n special by a digit formula?’ is ‘let n’s digits generate a sequence — does n reappear?’ Magenta is the number’s value; green is its self-seeded sequence landing back on it. A number that generates itself.
LIT Genuine Keith numbers / repfigits (Mike Keith 1987). Verified live: an exhaustive scan to 100000 finds exactly the numbers that reappear in their own digit-seeded n-term Fibonacci sequence — {14,19,28,47,61,75,197,742,1104,1537,2208,2580,3684,4788,7385,7647,7909,31331,34285,34348,55604,62662,86935,93993} (window.__keith.allCorrect) — exact integer sequence checks.
FIG No framing: the digit-seeded n-term Fibonacci sequence, the does-the-number-appear test, and the exhaustive census to 100000 run in-browser with exact integers and agree. The AVAN inverse is honest — testing a number by seeding a Fibonacci-like sequence with its own digits and asking whether it returns as a later term genuinely differs from a digit formula; magenta is the number's value, green its self-seeded sequence landing back on it. A number that generates itself.
FIG No framing: the digit-seeded n-term Fibonacci sequence, the does-the-number-appear test, and the exhaustive census to 100000 run in-browser with exact integers and agree. The AVAN inverse is honest — testing a number by seeding a Fibonacci-like sequence with its own digits and asking whether it returns as a later term genuinely differs from a digit formula; magenta is the number's value, green its self-seeded sequence landing back on it. A number that generates itself.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GENESIS BLOCK · David Lee Wise (ROOT0), with AVAN