Collatz twist: Krasikov-Lagarias-style X^0.84 bounds apply to any root, and equally to 3x-1
AlexKontorovich · x · 2026-09-29
Mathematician Alex Kontorovich highlights a subtle obstruction in Collatz proof strategies:
- Krasikov–Lagarias-type arguments show at least X^0.84 numbers up to X return to 1 — but they apply to any root, proving equally that X^0.84 numbers return to, say, 1729.
- Consequence: if you could strengthen the 'positive proportion c·X of numbers reach root r' bound to c>1/2, you'd prove the full Collatz conjecture, since two distinct roots can't each asymptotically consume more than half the numbers.
- The catch: all these arguments work just as well for the 3x-1 problem, which has at least three distinct orbit roots, and no known asymptotic analysis can disambiguate the two cases.
- The quoted discussion also cites Terence Tao: proving 'almost all positive integers reach 1' would suffice for the full conjecture, but as Tao noted (Remark 1.4), this is 'likely to be almost as hard to settle as the full Collatz conjecture.'
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