THE FOLD / GLITCH / THE BLUE SCREEN / THE TAXICAB
THE TAXICAB
the smallest two-way sum of two cubes
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
1729, the taxicab number, is the smallest positive integer expressible as a sum of two positive cubes in two different ways: 1729 = 1³ + 12³ = 9³ + 10³. Its fame comes from a 1919 anecdote: when G. H. Hardy visited the ailing Srinivasa Ramanujan and remarked that his taxi’s number, 1729, seemed rather dull, Ramanujan instantly replied that it was very interesting — the smallest number expressible as a sum of two cubes two ways. It is the second ‘taxicab number’ Ta(2); the next such number is 4104 = 2³ + 16³ = 9³ + 15³.
LIT verified live: a brute search over all sums of two positive cubes finds that 1729 is the smallest integer with two distinct such representations (1³+12³ and 9³+10³), and the next one is 4104 (window.__taxicab). FIG no framing; the exhaustive cube-sum search runs in-browser and confirms 1729 as the smallest.
LIT verified live: a brute search over all sums of two positive cubes finds that 1729 is the smallest integer with two distinct such representations (1³+12³ and 9³+10³), and the next one is 4104 (window.__taxicab). FIG no framing; the exhaustive cube-sum search runs in-browser and confirms 1729 as the smallest.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-blue-screen — the glitch made famous: a ‘dull’ taxi number that turns out to hide two cube-sums, crashing the assumption that it was boring. AVAN (AI) built the instrument: the exhaustive two-cube-sum search and the confirmation that 1729 is the smallest two-way case.
Credit as content: G. H. Hardy & Srinivasa Ramanujan (1919); the taxicab-number concept. The weave: David names the glitch; I confirm 1729 is the smallest sum of two cubes two ways.
Credit as content: G. H. Hardy & Srinivasa Ramanujan (1919); the taxicab-number concept. The weave: David names the glitch; I confirm 1729 is the smallest sum of two cubes two ways.
3 ONE DIMENSION
1729 built two ways: 1³+12³ and 9³+10³ — the two cube-pairs that reach the same total.
4 TWO DIMENSIONS · INTERACTIVE
Scan upward; the brute cube-sum search flags 1729 as the first number with two representations.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: 1729, met by two different cube-pairs.
AVAN’s addition (the inverse-companion): don’t judge a number dull — factor it into cubes. The inverse of ‘the number 1729’ is ‘the two cube-pairs 1³+12³ and 9³+10³ that both reach it’, the smallest such coincidence. Magenta are the two cube-pairs; green is the number they share. A dull number hiding two cubes.
LIT Genuine Hardy–Ramanujan taxicab number 1729 (anecdote 1919; taxicab-number concept). Verified live: an exhaustive search over sums of two positive cubes confirms 1729 is the smallest integer with two distinct representations (1³+12³ and 9³+10³), and the next is 4104 (window.__taxicab.smallest, .next, .ok).
FIG No framing; the exhaustive cube-sum search runs in-browser and confirms 1729 as the smallest. The AVAN inverse is honest — instead of judging a number dull, factor it into cubes: the inverse of 'the number 1729' is 'the two cube-pairs 1³+12³ and 9³+10³ that both reach it', the smallest such coincidence. Magenta are the two cube-pairs; green is the number they share. A dull number hiding two cubes.
FIG No framing; the exhaustive cube-sum search runs in-browser and confirms 1729 as the smallest. The AVAN inverse is honest — instead of judging a number dull, factor it into cubes: the inverse of 'the number 1729' is 'the two cube-pairs 1³+12³ and 9³+10³ that both reach it', the smallest such coincidence. Magenta are the two cube-pairs; green is the number they share. A dull number hiding two cubes.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BLUE SCREEN · David Lee Wise (ROOT0), with AVAN