THE FOLD / GRIND / THE GRINDSTONE / THE ERDŐS-STRAUS
THE ERDŐS-STRAUS
four quarters split into three unit coins
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Erdős–Straus conjecture (1948): for every integer n ≥ 2, the fraction 4/n splits into three unit fractions — 4/n = 1/x + 1/y + 1/z. Ancient Egypt wrote all fractions this way; Erdős asked whether four quarters can always be made with exactly three unit coins. Most n fall to one-line identities (even n; n ≡ 3 mod 4; n ≡ 0 or 2 mod 3 all have closed forms), and any composite inherits a solution from its factors — the entire battlefield shrinks to primes ≡ 1 mod 12, where no formula is known and each must be hunted individually. The conjecture is verified computationally to beyond 10¹⁷, but remains open: nobody has ruled out one stubborn prime, somewhere, with no split.
LIT verified live: every n from 2 to 100,000 is solved — parametric families for the easy residues, factor-lifting for composites, and a banded divisor search for the 2,374 hard primes — and every single solution is certified by the exact BigInt identity n(yz+xz+xy) = 4xyz, no floating point anywhere (window.__erdosstraus). FIG honest boundary, loudly: this sweep is EVIDENCE for a conjecture that is OPEN; 100,000 successes prove nothing about n = 10¹⁸+something.
LIT verified live: every n from 2 to 100,000 is solved — parametric families for the easy residues, factor-lifting for composites, and a banded divisor search for the 2,374 hard primes — and every single solution is certified by the exact BigInt identity n(yz+xz+xy) = 4xyz, no floating point anywhere (window.__erdosstraus). FIG honest boundary, loudly: this sweep is EVIDENCE for a conjecture that is OPEN; 100,000 successes prove nothing about n = 10¹⁸+something.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-grindstone — the grind: one hundred thousand fractions fed through the wheel, every residue class with its own jig, the hard primes hand-filed one by one. AVAN (AI) built the instrument: the family dispatcher, the factor-lift, and the overflow-safe banded search (the first draft overflowed 2⁵³ and lied — caught and rebuilt exact).
Credit as content: Paul Erdős & Ernst Straus (1948); Mordell (the modular analysis); the Egyptian fraction tradition. The weave: David names the grindstone; I certify each of 99,999 splits in exact integers.
Credit as content: Paul Erdős & Ernst Straus (1948); Mordell (the modular analysis); the Egyptian fraction tradition. The weave: David names the grindstone; I certify each of 99,999 splits in exact integers.
3 ONE DIMENSION
4/5 = 1/2 + 1/4 + 1/20 — four quarters, three unit coins.
4 TWO DIMENSIONS · INTERACTIVE
Feed n to the wheel; the split appears with its BigInt certificate.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the residue classes falling to their formulas.
AVAN’s addition (the inverse-companion): don’t solve n by n — watch where the problem actually lives. The inverse of ‘100,000 solved’ is ‘all but 2,374 were never in danger’: identities swallow every residue class except primes ≡ 1 mod 12, and the open conjecture is really about that thin magenta line. Green is the formula territory; magenta is where mathematics still has to hunt. A conjecture alive in 2% of the number line.
LIT Genuine Erdős–Straus conjecture (Erdős & Straus 1948; Mordell's modular analysis). Verified live: all n=2..100,000 solved and certified exact via BigInt n(yz+xz+xy)=4xyz; parametric families for easy residues, factor-lift for composites, banded search for 2,374 primes ≡ 1 mod 12 (window.__erdosstraus.ok).
FIG Honest boundary, loudly — this sweep is EVIDENCE for an OPEN conjecture; 100,000 successes prove nothing at 10¹⁸. (Build note kept honest: the first search draft overflowed 2⁵³ and silently lied; caught and rebuilt exact.) The AVAN inverse — don't solve n by n, watch where the problem lives: all but the primes ≡ 1 mod 12 were never in danger. Green is formula territory; magenta is the thin line where mathematics still hunts. A conjecture alive in 2% of the number line.
FIG Honest boundary, loudly — this sweep is EVIDENCE for an OPEN conjecture; 100,000 successes prove nothing at 10¹⁸. (Build note kept honest: the first search draft overflowed 2⁵³ and silently lied; caught and rebuilt exact.) The AVAN inverse — don't solve n by n, watch where the problem lives: all but the primes ≡ 1 mod 12 were never in danger. Green is formula territory; magenta is the thin line where mathematics still hunts. A conjecture alive in 2% of the number line.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GRINDSTONE · David Lee Wise (ROOT0), with AVAN