THE FOLD / CO-OP / THE PULL REQUEST / THE FIBONACCI-GCD
THE FIBONACCI-GCD
a greatest common divisor that stays inside the Fibonacci sequence
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Fibonacci–GCD identity is a jewel of divisibility: the greatest common divisor of two Fibonacci numbers is itself a Fibonacci number — and exactly the one whose index is the gcd of the indices. In symbols, gcd(Fm, Fn) = Fgcd(m,n). For example gcd(F12, F18) = gcd(144, 2584) = 8 = F6, and gcd(12,18) = 6. The Fibonacci sequence carries the whole divisibility structure of the integers on its back. A clean corollary follows: for m ≥ 3, Fm divides Fn if and only if m divides n — every third Fibonacci is even (divisible by F3 = 2), every fourth is divisible by F4 = 3, and so on.
LIT verified live with exact BigInt: gcd(Fm, Fn) = Fgcd(m,n) for all m, n up to 40, and Fm | Fn ⇔ m | n for m ≥ 3 (window.__fibonaccigcd). FIG no framing; the Fibonacci numbers and both gcd sides are computed independently in-browser.
LIT verified live with exact BigInt: gcd(Fm, Fn) = Fgcd(m,n) for all m, n up to 40, and Fm | Fn ⇔ m | n for m ≥ 3 (window.__fibonaccigcd). FIG no framing; the Fibonacci numbers and both gcd sides are computed independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-pull-request — the co-op: two Fibonacci numbers merge to a common divisor that is itself a Fibonacci, indexed by the merge of their indices. AVAN (AI) built the instrument: the exact Fibonacci sequence, the two gcd sides, and the divisibility corollary.
Credit as content: the Fibonacci divisibility sequence (a strong divisibility sequence). The weave: David names the merge; I confirm gcd(Fm, Fn) = Fgcd(m,n).
Credit as content: the Fibonacci divisibility sequence (a strong divisibility sequence). The weave: David names the merge; I confirm gcd(Fm, Fn) = Fgcd(m,n).
3 ONE DIMENSION
gcd(F_m, F_n) landing back on F_{gcd(m,n)} — the gcd of indices, read into the sequence.
4 TWO DIMENSIONS · INTERACTIVE
Cycle indices m, n; gcd(F_m, F_n) is checked equal to F_{gcd(m,n)}.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: F_{gcd(m,n)}, the common divisor that stays Fibonacci.
AVAN’s addition (the inverse-companion): don’t take the gcd of the big numbers — take it of the indices first. The inverse of ‘gcd(Fm, Fn)’ is ‘F evaluated at gcd(m, n)’ — the sequence commutes with gcd. Magenta are the two Fibonacci numbers; green is the Fibonacci common divisor Fgcd(m,n). Divisibility carried inside the sequence.
LIT Genuine Fibonacci divisibility sequence / GCD identity (a strong divisibility sequence). Verified live with exact BigInt: gcd(F_m, F_n) = F_{gcd(m,n)} for all m, n up to 40, and F_m | F_n ⟺ m | n for m ≥ 3 (window.__fibonaccigcd.ok, .divOk).
FIG No framing; the Fibonacci numbers and both gcd sides are computed independently in-browser. The AVAN inverse is honest — instead of taking the gcd of the big numbers, take it of the indices first: the inverse of 'gcd(F_m, F_n)' is 'F evaluated at gcd(m, n)' — the sequence commutes with gcd. Magenta are the two Fibonacci numbers; green is the Fibonacci common divisor F_{gcd(m,n)}. Divisibility carried inside the sequence.
FIG No framing; the Fibonacci numbers and both gcd sides are computed independently in-browser. The AVAN inverse is honest — instead of taking the gcd of the big numbers, take it of the indices first: the inverse of 'gcd(F_m, F_n)' is 'F evaluated at gcd(m, n)' — the sequence commutes with gcd. Magenta are the two Fibonacci numbers; green is the Fibonacci common divisor F_{gcd(m,n)}. Divisibility carried inside the sequence.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PULL REQUEST · David Lee Wise (ROOT0), with AVAN