THE FOLD / SPAWN / NULL ISLAND / THE BERTRAND POSTULATE
THE BERTRAND POSTULATE
a prime always between n and 2n
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Bertrand’s postulate guarantees primes never leave big gaps: for every integer n ≥ 1, there is at least one prime p with n < p ≤ 2n. Double any number and you are certain to have jumped over a prime. Joseph Bertrand conjectured it in 1845 and checked it up to three million; Chebyshev proved it in 1852, and Erdős gave a famously elegant elementary proof in 1932. It shows the primes, though irregular, are dense enough that they can never thin out to leave an interval [n, 2n] empty.
LIT verified live: a prime sieve confirms that for every n from 1 to 20000 there is a prime strictly greater than n and at most 2n; for n ≥ 2 the least such prime is strictly less than 2n (the only equality is n=1, where the prime is 2 = 2·1) (window.__bertrand). FIG no framing; the sieve and the interval check both run in-browser.
LIT verified live: a prime sieve confirms that for every n from 1 to 20000 there is a prime strictly greater than n and at most 2n; for n ≥ 2 the least such prime is strictly less than 2n (the only equality is n=1, where the prime is 2 = 2·1) (window.__bertrand). FIG no framing; the sieve and the interval check both run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at null-island — the spawn point: pick any n, double it, and a prime is guaranteed to have spawned somewhere in between. AVAN (AI) built the instrument: the prime sieve, the interval (n, 2n] check, and the least-prime ratio.
Credit as content: Joseph Bertrand (1845); Pafnuty Chebyshev (proof, 1852); Paul Erdős (elementary proof, 1932). The weave: David names the spawn; I confirm a prime always lies in (n, 2n].
Credit as content: Joseph Bertrand (1845); Pafnuty Chebyshev (proof, 1852); Paul Erdős (elementary proof, 1932). The weave: David names the spawn; I confirm a prime always lies in (n, 2n].
3 ONE DIMENSION
The interval (n, 2n] on the number line, with the prime(s) inside it highlighted — always at least one.
4 TWO DIMENSIONS · INTERACTIVE
Cycle n; the least prime in (n, 2n] is shown, always present, and its ratio to n stays below 2.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the prime always waiting in (n, 2n].
AVAN’s addition (the inverse-companion): don’t hunt for a prime — double and it’s there. The inverse of ‘is there a prime near n?’ is ‘yes — somewhere in (n, 2n], always’, so the primes never leave a doubling-gap empty. Magenta is the interval (n, 2n]; green are the primes guaranteed inside it. Primes that never leave a gap.
LIT Genuine Bertrand's postulate (Joseph Bertrand 1845; Chebyshev's proof 1852; Erdős's elementary proof 1932). Verified live: a sieve confirms a prime in (n, 2n] for every n from 1 to 10000+; for n≥2 the least such prime is strictly < 2n (worst ratio ~1.67), with n=1 the only equality (p=2=2·1) (window.__bertrand.ok, .worst2, .LIM).
FIG No framing; the sieve and the interval check both run in-browser. The AVAN inverse is honest — instead of hunting for a prime, double and it's there: the inverse of 'is there a prime near n?' is 'yes — somewhere in (n, 2n], always', so the primes never leave a doubling-gap empty. Magenta is the interval (n, 2n]; green are the primes guaranteed inside it. Primes that never leave a gap.
FIG No framing; the sieve and the interval check both run in-browser. The AVAN inverse is honest — instead of hunting for a prime, double and it's there: the inverse of 'is there a prime near n?' is 'yes — somewhere in (n, 2n], always', so the primes never leave a doubling-gap empty. Magenta is the interval (n, 2n]; green are the primes guaranteed inside it. Primes that never leave a gap.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NULL ISLAND · David Lee Wise (ROOT0), with AVAN