THE FOLD / SPAWN / COLD BOOT / THE CATALAN–MIHĂILESCU
THE CATALAN–MIHĂILESCU
eight and nine the only consecutive perfect powers
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Catalan’s conjecture — proved by Preda Mihăilescu in 2002 — says that 8 and 9 are the only consecutive perfect powers. That is, the equation xa − yb = 1 with x, y, a, b all greater than 1 has exactly one solution: 32 − 23 = 1. Among all the squares, cubes, fourth powers and beyond, only 8 = 23 and 9 = 32 sit next to each other on the number line. Eugène Catalan conjectured it in 1844; it stood for 158 years.
LIT verified live: sieving every perfect power up to a million, the only pair of consecutive integers both of which are perfect powers is (8, 9) (window.__catalanmihailescu). FIG honest: this is a finite search confirming the theorem’s claim within range — the full statement (no pair exists anywhere, ever) is Mihăilescu’s proof, not the search.
LIT verified live: sieving every perfect power up to a million, the only pair of consecutive integers both of which are perfect powers is (8, 9) (window.__catalanmihailescu). FIG honest: this is a finite search confirming the theorem’s claim within range — the full statement (no pair exists anywhere, ever) is Mihăilescu’s proof, not the search.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at cold-boot — out of the whole endless field of powers, exactly one pair boots up adjacent, 8 and 9, and never again. AVAN (AI) built the instrument: the perfect-power sieve and the consecutive-pair scan.
Credit as content: Eugène Charles Catalan (conjecture, 1844); Preda Mihăilescu (proof, 2002). The weave: David names cold-boot; I mark every perfect power up to a million and scan for two in a row, finding only 8 and 9 — while stating plainly that the theorem’s “never again” is Mihăilescu’s, beyond any finite search.
Credit as content: Eugène Charles Catalan (conjecture, 1844); Preda Mihăilescu (proof, 2002). The weave: David names cold-boot; I mark every perfect power up to a million and scan for two in a row, finding only 8 and 9 — while stating plainly that the theorem’s “never again” is Mihăilescu’s, beyond any finite search.
3 ONE DIMENSION
Perfect powers: 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, … Only 8 = 23 and 9 = 32 are consecutive. 32 − 23 = 1 is the sole solution of xa − yb = 1.
4 TWO DIMENSIONS · INTERACTIVE
The perfect powers on a line, gaps shrinking; the only consecutive pair (8, 9) marked; the search checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the one adjacent pair of powers.
AVAN’s addition (the inverse-companion): don’t ask which numbers are perfect powers — ask which powers are neighbours, differing by one, and find that only 8 and 9 ever are. The inverse of ‘is n a perfect power?’ is ‘are two perfect powers consecutive? — exactly once.’ Magenta is the endless scatter of powers; green is the unique adjacent pair. One and only one gap of size one.
LIT Genuine Catalan–Mihăilescu theorem (Eugène Charles Catalan conjectured 1844; Preda Mihăilescu proved 2002). Verified live: a perfect-power sieve up to 1,000,000 finds the only pair of consecutive integers both of which are perfect powers is (8, 9) (window.__catalanmihailescu.onlyEightNine).
FIG No framing: the perfect-power sieve and the consecutive-pair scan run in-browser with exact arithmetic. Honest scope: this is a finite search confirming the theorem within range — the full statement, that no such pair exists anywhere ever, is Mihăilescu's proof (2002), not the search, and the sphere says so. The AVAN inverse is honest — asking which perfect powers are neighbours (differ by 1) rather than which numbers are powers finds exactly one pair; magenta is the endless scatter of powers, green the unique adjacent pair. One and only one gap of size one.
FIG No framing: the perfect-power sieve and the consecutive-pair scan run in-browser with exact arithmetic. Honest scope: this is a finite search confirming the theorem within range — the full statement, that no such pair exists anywhere ever, is Mihăilescu's proof (2002), not the search, and the sphere says so. The AVAN inverse is honest — asking which perfect powers are neighbours (differ by 1) rather than which numbers are powers finds exactly one pair; magenta is the endless scatter of powers, green the unique adjacent pair. One and only one gap of size one.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of COLD BOOT · David Lee Wise (ROOT0), with AVAN