THE FOLD / LOOT / THE HOARD / THE THABIT
THE THABIT
two numbers each the sum of the other's divisors
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Amicable numbers are two different numbers where each equals the sum of the other’s proper divisors. The classic pair is (220, 284): the divisors of 220 sum to 284, and the divisors of 284 sum to 220. In the 9th century Thabit ibn Qurra found a formula that spins such pairs out of primes: for n≥2, if p = 3·2n-1-1, q = 3·2n-1, and r = 9·22n-1-1 are all prime, then 2n·p·q and 2n·r are amicable. The primes align rarely — only n = 2, 4, 7 work below n = 8 — which is why amicable pairs are scarce and prized.
LIT verified live: Thabit’s rule at n = 2, 4, 7 yields (220,284), (17296,18416), (9363584,9437056), and each pair is confirmed amicable by directly summing proper divisors (σ*(A)=B and σ*(B)=A); the classic pair and the perfect-number sanity check (σ*(6)=6) also hold (window.__thabit). FIG no framing; the primality tests, the rule, and the divisor sums all run in-browser.
LIT verified live: Thabit’s rule at n = 2, 4, 7 yields (220,284), (17296,18416), (9363584,9437056), and each pair is confirmed amicable by directly summing proper divisors (σ*(A)=B and σ*(B)=A); the classic pair and the perfect-number sanity check (σ*(6)=6) also hold (window.__thabit). FIG no framing; the primality tests, the rule, and the divisor sums all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-hoard — a paired treasure: two numbers that each hold exactly the other’s worth, a friendship measured in divisors. AVAN (AI) built the instrument: the proper-divisor sum, the primality test, Thabit’s p/q/r rule, and the amicability check.
Credit as content: Thabit ibn Qurra (9th c.); the n=4 and n=7 pairs later found by Fermat and Descartes. The weave: David names the hoard; I confirm the rule produces genuinely amicable pairs, checked by summing divisors.
Credit as content: Thabit ibn Qurra (9th c.); the n=4 and n=7 pairs later found by Fermat and Descartes. The weave: David names the hoard; I confirm the rule produces genuinely amicable pairs, checked by summing divisors.
3 ONE DIMENSION
The proper divisors of 220 sum to 284 (green bars); the proper divisors of 284 sum to 220 (blue bars).
4 TWO DIMENSIONS · INTERACTIVE
Cycle through the working n; see p,q,r come out prime and the resulting pair confirmed amicable by divisor sums.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the amicable pair, each pointing to the other.
AVAN’s addition (the inverse-companion): don’t sum a number’s divisors to itself — sum them to its partner. The inverse of ‘σ*(A)=B’ is ‘σ*(B)=A’: apply the divisor-sum twice and you return to the start. Magenta is the perfect number (6, 28) — amicable with itself, the fixed point σ*(n)=n. Friendship as a two-step return.
LIT Genuine Thabit ibn Qurra amicable-number rule (9th c.); the n=4 and n=7 pairs later rediscovered by Fermat and Descartes. Verified live: the rule at n=2,4,7 yields (220,284), (17296,18416), (9363584,9437056), each confirmed amicable by directly summing proper divisors (σ*(A)=B and σ*(B)=A); the classic pair and the perfect-number sanity σ*(6)=6 also hold (window.__thabit.ruleAmicable, .classic, .perfect6).
FIG No framing; the primality tests, the rule, and the divisor sums all run in-browser. The AVAN inverse is honest — instead of summing a number's divisors to itself, sum them to its partner: apply the divisor-sum twice and you return to the start. Magenta is the perfect number (6, 28) — amicable with itself, the fixed point σ*(n)=n. Friendship as a two-step return.
FIG No framing; the primality tests, the rule, and the divisor sums all run in-browser. The AVAN inverse is honest — instead of summing a number's divisors to itself, sum them to its partner: apply the divisor-sum twice and you return to the start. Magenta is the perfect number (6, 28) — amicable with itself, the fixed point σ*(n)=n. Friendship as a two-step return.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HOARD · David Lee Wise (ROOT0), with AVAN