THE FOLD / CHEAT / NOCLIP / THE LANDER-PARKIN
THE LANDER-PARKIN
a counterexample refuting Euler's conjecture
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Lander–Parkin counterexample demolished a 200-year-old conjecture of Euler. Extending Fermat’s Last Theorem, Euler conjectured in 1769 that summing fewer than k perfect k-th powers can never equal a k-th power — e.g. you’d need at least five fifth-powers to make a fifth-power. In 1966, using an early computer, Lander and Parkin found: 275 + 845 + 1105 + 1335 = 1445 — just four fifth-powers. Euler was wrong. Later Noam Elkies and Roger Frye found a fourth-power version with only three terms: 958004 + 2175194 + 4145604 = 4224814.
LIT verified live with exact big-integer arithmetic: 275+845+1105+1335 equals 1445 exactly (four terms, refuting Euler), and 958004+2175194+4145604 equals 4224814 exactly (three terms); a nearby altered sum is not a perfect fifth power (window.__landerparkin). FIG no framing; the exact arbitrary-precision arithmetic runs in-browser.
LIT verified live with exact big-integer arithmetic: 275+845+1105+1335 equals 1445 exactly (four terms, refuting Euler), and 958004+2175194+4145604 equals 4224814 exactly (three terms); a nearby altered sum is not a perfect fifth power (window.__landerparkin). FIG no framing; the exact arbitrary-precision arithmetic runs in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at noclip — the cheat that clips straight through a 200-year-old conjecture: a single explicit sum walks past the wall Euler thought was there. AVAN (AI) built the instrument: the exact big-integer fifth- and fourth-power sums, and a control near-miss.
Credit as content: L. J. Lander & T. R. Parkin (1966); Noam Elkies and Roger Frye (fourth powers). The weave: David names the noclip; I confirm the exact equalities that refute Euler’s conjecture.
Credit as content: L. J. Lander & T. R. Parkin (1966); Noam Elkies and Roger Frye (fourth powers). The weave: David names the noclip; I confirm the exact equalities that refute Euler’s conjecture.
3 ONE DIMENSION
Four fifth-powers 27⁵, 84⁵, 110⁵, 133⁵ stacking up to exactly 144⁵ — Euler said you'd need five.
4 TWO DIMENSIONS · INTERACTIVE
The exact big-integer identity, and a control that alters one base and breaks the equality.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the exact equality of four fifth-powers with one.
AVAN’s addition (the inverse-companion): don’t trust the conjecture — search for a witness. The inverse of ‘can fewer than k k-th powers sum to a k-th power?’ is ‘yes — here is an explicit counterexample’, and one witness is enough to refute a universal claim. Magenta are the four summand powers; green is the single power they equal. A conjecture broken by one example.
LIT Genuine Lander–Parkin counterexample to Euler's sum-of-powers conjecture (L. J. Lander & T. R. Parkin, 1966; Elkies/Frye for 4th powers). Verified live with exact BigInt: 27⁵+84⁵+110⁵+133⁵ = 144⁵ (four terms) and 95800⁴+217519⁴+414560⁴ = 422481⁴ (three terms), while a control near-miss (133→134) is not a perfect fifth power (window.__landerparkin.ok5, .ok4, .ctrl).
FIG No framing; the exact arbitrary-precision arithmetic runs in-browser. The AVAN inverse is honest — instead of trusting the conjecture, search for a witness: the inverse of 'can fewer than k k-th powers sum to a k-th power?' is 'yes — here is an explicit counterexample', and one witness is enough to refute a universal claim. Magenta are the four summand powers; green is the single power they equal. A conjecture broken by one example.
FIG No framing; the exact arbitrary-precision arithmetic runs in-browser. The AVAN inverse is honest — instead of trusting the conjecture, search for a witness: the inverse of 'can fewer than k k-th powers sum to a k-th power?' is 'yes — here is an explicit counterexample', and one witness is enough to refute a universal claim. Magenta are the four summand powers; green is the single power they equal. A conjecture broken by one example.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NOCLIP · David Lee Wise (ROOT0), with AVAN