THE FOLD / GLITCH / DIVIDE BY ZERO / THE KUMMER
THE KUMMER
count the carries to know how many times p divides
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Kummer’s theorem reveals a hidden bridge between addition and divisibility: the number of times a prime p divides the binomial coefficient C(m+n, n) equals exactly the number of carries when you add m and n in base p. A dry counting question — how divisible is this binomial? — is answered by the schoolyard mechanics of carrying digits. No factorials need to be computed; just add in base p and count the carries.
LIT verified live: over 20000 random (m, n, p), the p-adic valuation of C(m+n, n) — computed by Legendre’s formula — equals the carry count of m+n in base p (window.__kummer). FIG no framing; exact integer arithmetic, no big numbers.
LIT verified live: over 20000 random (m, n, p), the p-adic valuation of C(m+n, n) — computed by Legendre’s formula — equals the carry count of m+n in base p (window.__kummer). FIG no framing; exact integer arithmetic, no big numbers.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at divide-by-zero — the domain of divisibility’s edge; Kummer says precisely how many times p goes into a binomial, and it does so by counting carries, not by dividing. AVAN (AI) built the instrument: Legendre’s valuation formula, the base-p carry counter, and the equality check.
Credit as content: Ernst Kummer (1852). The weave: David names divide-by-zero; I compute how many times p divides C(m+n, n) two ways — by Legendre’s prime-power formula and by counting carries in base-p addition — and confirm they always agree, addition and divisibility revealed as one.
Credit as content: Ernst Kummer (1852). The weave: David names divide-by-zero; I compute how many times p divides C(m+n, n) two ways — by Legendre’s prime-power formula and by counting carries in base-p addition — and confirm they always agree, addition and divisibility revealed as one.
3 ONE DIMENSION
Add m and n in base p; each place that overflows is a carry. The count of carries is exactly the power of p dividing C(m+n, n). Divisibility, read off an addition.
4 TWO DIMENSIONS · INTERACTIVE
Pick m, n, p; see the base-p addition with its carries, and the p-adic valuation of the binomial, checked equal.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: divisibility counted as carries.
AVAN’s addition (the inverse-companion): find how many times a prime divides a binomial not by factoring the factorials but by counting carries in one base-p addition. The inverse of ‘compute C(m+n,n) and factor out p’ is ‘add m and n in base p — the carries are the exponent.’ Magenta is the factorial factorization avoided; green is the carry count. Divisibility hiding inside addition.
LIT Genuine Kummer's theorem (Ernst Kummer 1852). Verified live: over 20000 random (m,n,p) with p prime, the exponent of p in C(m+n,n) computed by Legendre's formula v_p(k!)=Σ⌊k/p^i⌋ equals the number of carries in the base-p addition of m and n (window.__kummer.kummerHolds) — exact integer arithmetic, no large numbers formed.
FIG No framing: Legendre's valuation formula, the base-p carry counter, and the equality check run in-browser with exact integers and agree. The AVAN inverse is honest — finding how many times p divides a binomial by counting carries in one base-p addition genuinely replaces factoring the factorials; magenta is that factorization avoided, green the carry count. Divisibility hiding inside addition.
FIG No framing: Legendre's valuation formula, the base-p carry counter, and the equality check run in-browser with exact integers and agree. The AVAN inverse is honest — finding how many times p divides a binomial by counting carries in one base-p addition genuinely replaces factoring the factorials; magenta is that factorization avoided, green the carry count. Divisibility hiding inside addition.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of DIVIDE BY ZERO · David Lee Wise (ROOT0), with AVAN