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THE VON STAUDT-CLAUSEN
a Bernoulli denominator read off from primes
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The von Staudt–Clausen theorem reveals the exact denominator of every Bernoulli number. The Bernoulli numbers B2n are wild rationals with enormous numerators — yet their denominators are astonishingly simple: the denominator of B2n is precisely the product of the primes p for which (p−1) divides 2n. So denom(B2)=6=2·3, denom(B10)=66=2·3·11, and 2 and 3 divide every even-index Bernoulli denominator (since p−1∈{1,2} always divides 2n). A messy fraction’s bottom half is read straight off a divisibility condition on primes.
LIT verified live: computing the Bernoulli numbers exactly as reduced fractions, the denominator of B2n equals ∏(p−1)|2n p for every n from 1 to 15 (window.__von_staudt). FIG no framing; the exact-fraction Bernoulli recurrence and the prime product run in-browser.
LIT verified live: computing the Bernoulli numbers exactly as reduced fractions, the denominator of B2n equals ∏(p−1)|2n p for every n from 1 to 15 (window.__von_staudt). FIG no framing; the exact-fraction Bernoulli recurrence and the prime product run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-vault — a fraction whose denominator is a locked product of primes, opened by one divisibility rule. AVAN (AI) built the instrument: the exact rational Bernoulli recurrence (BigInt fractions), the divisor-prime product, and the denominator match.
Credit as content: Karl von Staudt & Thomas Clausen (independently, 1840). The weave: David names the vault; I confirm each Bernoulli denominator is exactly the product of the primes p with (p−1)|2n.
Credit as content: Karl von Staudt & Thomas Clausen (independently, 1840). The weave: David names the vault; I confirm each Bernoulli denominator is exactly the product of the primes p with (p−1)|2n.
3 ONE DIMENSION
B_2n as a reduced fraction; its denominator is the product of primes p where (p−1) divides 2n.
4 TWO DIMENSIONS · INTERACTIVE
Pick n; see B_2n, the primes p with (p−1)|2n, and their product — exactly the denominator.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the denominator of B_2n.
AVAN’s addition (the inverse-companion): don’t reduce the fraction — read the primes. The inverse of ‘compute B2n and simplify’ is ‘its denominator is ∏ p over primes with (p−1)|2n.’ Magenta are the qualifying primes; green is their product, the denominator. The bottom is written in primes.
LIT Genuine von Staudt–Clausen theorem (Karl von Staudt & Thomas Clausen, independently 1840). Verified live: computing Bernoulli numbers exactly as reduced BigInt fractions via the recurrence, the denominator of B_2n equals ∏ of primes p with (p−1)|2n for every n=1..15 (e.g. denom(B_10)=66=2·3·11) (window.__von_staudt.matches).
FIG No framing: the exact-fraction Bernoulli recurrence and the prime product run in-browser. The AVAN inverse is honest — instead of computing B_2n and reducing the fraction, its denominator is read directly from a divisibility rule: ∏ p over primes with (p−1)|2n. Magenta are the qualifying primes; green is their product, the denominator. The bottom is written in primes.
FIG No framing: the exact-fraction Bernoulli recurrence and the prime product run in-browser. The AVAN inverse is honest — instead of computing B_2n and reducing the fraction, its denominator is read directly from a divisibility rule: ∏ p over primes with (p−1)|2n. Magenta are the qualifying primes; green is their product, the denominator. The bottom is written in primes.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE-VAULT · David Lee Wise (ROOT0), with AVAN