THE FOLD / GLITCH / DIVIDE BY ZERO / THE RAMANUJAN CONGRUENCE
THE RAMANUJAN CONGRUENCE
hidden divisibilities in the partition numbers
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Ramanujan’s congruences are astonishing hidden divisibilities in the partition numbers p(n) — the count of ways to write n as a sum of positive integers. Ramanujan noticed, from a hand-written table, three exact patterns: p(5n + 4) ≡ 0 (mod 5), p(7n + 5) ≡ 0 (mod 7), and p(11n + 6) ≡ 0 (mod 11). Every fifth partition number from p(4) is divisible by 5; every seventh from p(5) by 7; every eleventh from p(6) by 11. There is no such simple congruence for any other prime — 5, 7, 11 are special.
LIT verified live (exact BigInt): computing p(n) by Euler’s pentagonal recurrence, p(5n+4) is divisible by 5, p(7n+5) by 7, and p(11n+6) by 11 for every n in range (window.__ramanujan). FIG no framing; exact big-integer partition counts checked against the three moduli.
LIT verified live (exact BigInt): computing p(n) by Euler’s pentagonal recurrence, p(5n+4) is divisible by 5, p(7n+5) by 7, and p(11n+6) by 11 for every n in range (window.__ramanujan). FIG no framing; exact big-integer partition counts checked against the three moduli.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at divide-by-zero — whole arithmetic progressions of partition numbers that come out exactly zero modulo 5, 7, 11, with no obvious reason in the definition. That unexpected vanishing is the mechanic. AVAN (AI) built the instrument: the pentagonal-recurrence partition counter in big integers, and the three congruence checks.
Credit as content: Srinivasa Ramanujan (1919). The weave: David names divide-by-zero; I compute p(n) exactly by adding and subtracting earlier partition counts at the generalized pentagonal offsets, then confirm that p(5n+4), p(7n+5), p(11n+6) vanish modulo 5, 7, 11 — the congruences Ramanujan saw in a table.
Credit as content: Srinivasa Ramanujan (1919). The weave: David names divide-by-zero; I compute p(n) exactly by adding and subtracting earlier partition counts at the generalized pentagonal offsets, then confirm that p(5n+4), p(7n+5), p(11n+6) vanish modulo 5, 7, 11 — the congruences Ramanujan saw in a table.
3 ONE DIMENSION
p(4)=5, p(9)=30, p(14)=135, p(19)=490 — all divisible by 5 (the 5n+4 class). p(5)=7, p(12)=77 by 7. p(6)=11, p(17)=297 by 11. No such rule for any other prime.
4 TWO DIMENSIONS · INTERACTIVE
Partition numbers mod 5, 7, 11; the arithmetic progressions that vanish highlighted; the congruences checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: whole progressions vanishing mod a prime.
AVAN’s addition (the inverse-companion): don’t just count partitions — watch them modulo a prime, and find entire arithmetic progressions that come out zero. The inverse of ‘how many partitions of n’ is ‘which residue classes of n force p(n) ≡ 0.’ Magenta is the raw partition count; green is the vanishing residue class mod 5, 7, or 11. Hidden zeros in the counting.
LIT Genuine Ramanujan congruences (Srinivasa Ramanujan, 1919). Verified live with exact BigInt: the partition function computed by Euler's pentagonal recurrence satisfies p(5n+4) ≡ 0 (mod 5) (window.__ramanujan.c5), p(7n+5) ≡ 0 (mod 7) (window.__ramanujan.c7), and p(11n+6) ≡ 0 (mod 11) (window.__ramanujan.c11) for every n with the argument up to 600.
FIG No framing: the pentagonal-recurrence partition counter and the three congruence checks all run in-browser in exact big integers. The AVAN inverse is honest — reading the partition numbers modulo a prime and finding entire residue classes that vanish (rather than just counting partitions) is the genuine phenomenon Ramanujan spotted; magenta is the raw partition count, green the vanishing residue class mod 5, 7, or 11. Hidden zeros in the counting.
FIG No framing: the pentagonal-recurrence partition counter and the three congruence checks all run in-browser in exact big integers. The AVAN inverse is honest — reading the partition numbers modulo a prime and finding entire residue classes that vanish (rather than just counting partitions) is the genuine phenomenon Ramanujan spotted; magenta is the raw partition count, green the vanishing residue class mod 5, 7, or 11. Hidden zeros in the counting.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of DIVIDE BY ZERO · David Lee Wise (ROOT0), with AVAN