THE FOLD / BOSS / THE FINAL BOSS / THE MIHĂILESCU
THE MIHĂILESCU
the only two powers that touch
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
In 1844 Eugène Catalan mailed a one-paragraph conjecture to Crelle’s Journal: 8 and 9 are the only consecutive perfect powers — the only solution of xᵖ − yⁿ = 1 in integers greater than 1 is 3² − 2³. It stood for 158 years. Tijdeman (1976) proved the solutions were finite; the bounds were astronomically useless. Then in 2002 Preda Mihăilescu — working largely outside academia — killed it completely with a proof from the theory of cyclotomic fields, no computers involved. Among the infinite towers of squares, cubes, and higher powers scattered along the number line, exactly one pair of neighbours ever touch.
LIT verified live: every perfect power up to 10¹² is generated (1,010,195 of them), sorted, and scanned — exactly one consecutive pair exists: (8, 9) (window.__mihailescu). FIG honest boundary: the in-browser sweep verifies the theorem to 10¹²; the claim for ALL integers is Mihăilescu’s 2002 theorem, cited as the mountain it is — a computation can witness it, only the cyclotomic proof owns it.
LIT verified live: every perfect power up to 10¹² is generated (1,010,195 of them), sorted, and scanned — exactly one consecutive pair exists: (8, 9) (window.__mihailescu). FIG honest boundary: the in-browser sweep verifies the theorem to 10¹²; the claim for ALL integers is Mihăilescu’s 2002 theorem, cited as the mountain it is — a computation can witness it, only the cyclotomic proof owns it.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-final-boss — the boss: a one-line challenge that outlived every challenger for a century and a half, finally beaten by an outsider with a build nobody expected. AVAN (AI) built the instrument: the power-tower generator and the exhaustive adjacency scan.
Credit as content: Eugène Catalan (1844); Robert Tijdeman (1976); Preda Mihăilescu (2002). The weave: David names the final boss; I walk the first trillion integers and find the single touch.
Credit as content: Eugène Catalan (1844); Robert Tijdeman (1976); Preda Mihăilescu (2002). The weave: David names the final boss; I walk the first trillion integers and find the single touch.
3 ONE DIMENSION
The perfect powers below 300 on the line — and the one place two of them touch.
4 TWO DIMENSIONS · INTERACTIVE
Sweep the census; watch the gap between neighbouring powers grow — after 8,9 it never closes again.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the towers of powers rising side by side.
AVAN’s addition (the inverse-companion): don’t search for solutions — ask why the powers repel. The inverse of ‘8 and 9 touch’ is ‘everywhere else, a forced gap’: squares spread as 2n+1, cubes as 3n², and the arithmetic leaves no second seam. Magenta is the single point of contact, never repeated; green is the growing silence between towers. One touch in all of infinity — and it took 158 years to prove the silence.
LIT Genuine Catalan conjecture / Mihăilescu's theorem (Eugène Catalan 1844; Robert Tijdeman 1976; Preda Mihăilescu 2002). Verified live: every perfect power ≤ 10¹² (1,010,195 of them) generated and scanned — the only consecutive pair is (8,9) = (2³,3²) (window.__mihailescu.ok).
FIG Honest boundary — the sweep verifies to 10¹²; the claim for ALL integers is Mihăilescu's theorem, cited as the mountain it is: a computation can witness it, only the cyclotomic proof owns it. The AVAN inverse — don't search for solutions, ask why the powers repel: the inverse of '8 and 9 touch' is 'everywhere else, a forced gap'. Magenta is the single point of contact never repeated; green is the growing silence between towers. One touch in all of infinity — 158 years to prove the silence.
FIG Honest boundary — the sweep verifies to 10¹²; the claim for ALL integers is Mihăilescu's theorem, cited as the mountain it is: a computation can witness it, only the cyclotomic proof owns it. The AVAN inverse — don't search for solutions, ask why the powers repel: the inverse of '8 and 9 touch' is 'everywhere else, a forced gap'. Magenta is the single point of contact never repeated; green is the growing silence between towers. One touch in all of infinity — 158 years to prove the silence.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE FINAL BOSS · David Lee Wise (ROOT0), with AVAN