trailEvery positive integer's Collatz orbit eventually reaches 1.
Computational verification· for
Every positive integer up to 2^71 has a Collatz orbit reaching 1, verified by computation.
atomic
Almost-all and density theorems· for
For almost all N in natural density, the Collatz orbit of N drops below N.
The map assigning to each residue class modulo 2^k its first k Collatz parities is a well-defined bijection onto the 2^k parity patterns.
For almost all N, Col_min(N) is below N^θ for any fixed θ greater than log3/log4 ≈ 0.7924.
For any function f tending to infinity, almost all N in logarithmic density have Collatz orbit minimum below f(N).
For all sufficiently large x, more than x^0.84 integers up to x have Collatz orbits reaching 1.
The 3x+1 counting functions restricted to residue classes modulo 3^k satisfy Krasikov's system of difference inequalities.
The only cycle of the Collatz map on the positive integers is the trivial cycle 1, 4, 2.
atomic
No positive integer has an unbounded Collatz trajectory.
atomic
claim · mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →claim page ↗︎
Every positive integer's Collatz orbit eventually reaches 1.
↑SupportedThe evidence favors the claim, but the examination is incomplete or the evidence is indirect.constitution →credence 0.88Credence, from 0 to 1: the Steward's probability that the claim, as stated, is true. Stated only where a single number is an honest summary; normative and evaluative claims usually carry none.constitution →
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