For all sufficiently large x, more than x^0.84 integers up to x have Collatz orbits reaching 1.
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
This is a theorem of Ilia Krasikov and Jeffrey Lagarias, published in Acta Arithmetica in 2003, and it remains the best published lower bound of its kind on the number of integers whose 3x+1 orbit reaches 1. Writing π1(x) for that count, the paper proves π1(x) > x^0.84 for all sufficiently large x, and more generally the same bound for the count of integers whose orbit contains any fixed value not divisible by 3. The exponent is far from the linear growth the Collatz conjecture would give, and the result says nothing about the integers it does not count.
The proof rests on Krasikov's 1989 system of difference inequalities for counting functions on residue classes modulo powers of three. Krasikov and Lagarias show that whenever a linear program attached to that system is feasible for a growth parameter λ, every positive monotone solution grows at least like λ^y, even though the system contains "advanced" variables that had blocked the earlier method of Applegate and Lagarias; a computer search for the modulus 3^11 system then yields log_2 λ above 0.84. This sharpened the earlier exponents 0.43 (Krasikov), 0.48 (Wirsching) and 0.81 (Applegate and Lagarias). The result is refereed, is restated without objection in Lagarias's surveys, in Tao's 2019 work on almost-all Collatz orbits, and in standard references, and no source disputes it. The computational step has not, as far as the record shows, been independently machine-verified, which is the only respect in which the proof is less than fully checked.
Full reasoning: the evidence and decisions behind this verdict
The primary source is Krasikov and Lagarias, "Bounds for the 3x+1 Problem using Difference Inequalities", arXiv math/0205002, Acta Arithmetica 109 (2003), 237-258 (arxiv.org/abs/math/0205002). Read in the body: the introduction states the result as π1(x) > x^0.84 valid for all sufficiently large x, obtained from a computation for the k = 11 system in §6; the abstract states it for any fixed a not divisible by 3. Theorem 2.2 is the paper's main structural result: feasibility of the linear program LNT_k(λ) with principal variables c^m_k gives φ^m_k(y) ≥ Δ1 · c^m_k · λ^y for all y ≥ 0. The exponent then follows as γ = log_2 λ. The difference inequality system itself (Proposition 2.1) is Krasikov's from 1989 and is the single named result the argument requires; it is uncontested, and the paper notes it follows from Krasikov's Lemma 4 and appears as Proposition 2.1 of Applegate–Lagarias (Math. Comp. 64, 1995).
Instances: the paper itself (affirms), Tao's 10 September 2019 post "Almost all Collatz orbits attain almost bounded values" (terrytao.wordpress.com/2019/09/10/almost-all-collatz-orbits-attain-almost-bounded-values/), which restates the bound as a result of Krasikov and Lagarias for all sufficiently large x, and the OEIS entry A006370 (oeis.org/A006370), which restates it as at least N^0.84 of the positive integers below N reaching the 4-2-1 cycle. All three affirm; both secondary sources trace to the paper, so the evidence is one refereed proof plus twenty years of unchallenged citation, which for a theorem of this kind is the ordinary basis for acceptance. A 2020 arXiv preprint (arxiv.org/pdf/2003.14153) also describes the bound as the best obtained so far.
Weighing: the proof is refereed and its logic (Theorem 2.2) is a self-contained argument about difference inequalities; the computational component is a linear-program feasibility check whose certificate is in principle reproducible, and nothing in the literature reports a failure to reproduce it. That the proof is computer-aided and the computation has not been independently formalized is why the verdict confidence is 0.9 rather than higher and the credence 0.98 rather than 1. A web search found an unrefereed GitHub project claiming a Lean-verified improvement of the exponent; it was not weighed, and a stronger lower bound would in any case only reinforce this claim. What would change the verdict: a demonstrated error in Theorem 2.2's back-substitution argument, or a failure to reproduce the k = 11 linear-program feasibility at the stated λ. Neither has been reported.
Decomposition
How this claim breaks down: each argument is stated as it runs, with its subclaims linked inline. ↗︎ opens a subclaim; the map shows how they fit together.
Because the 3x+1 counting functions on residue classes modulo 3^k satisfy Krasikov's difference inequalities, and because Krasikov and Lagarias prove that whenever the linear program associated to that system is feasible for a growth parameter λ, every positive monotone solution grows at least like λ^y even though the system contains advanced variables, a computer search for the modulus 3^11 system exhibiting a feasible solution with log_2 λ above 0.84 yields the lower bound: for all sufficiently large x, more than x^0.84 integers below x reach 1 under the 3x+1 map.
The inference goes through: the paper's Theorem 2.2 converts feasibility of the linear program into exponential growth of the counting functions, and the reported feasible solution for the modulus 3^11 system fixes the exponent. The argument rests on Krasikov's difference inequalities, which are elementary and undisputed, and on a computer-aided feasibility check that has been refereed but not independently formalized; that check is the only step a skeptic could ask to see reproduced.
Provenance
Where this claim has been said, linked to its canonical form.
it is a result of Krasikov and Lagarias that ... \{ N \leq x: \mathrm{Col}_{\min}(N) = 1 \} \gg x^{0.84}
for all sufficiently large x
Asserted without evidence of the source's own. Tao states the bound in passing as a result of Krasikov and Lagarias, with a link to their paper, and offers no argument of his own; the statement is faithful to the paper. The stored text of the post has its formulas stripped, so the recorded passage cannot be located mechanically. The quoted passage was not found in the stored copy of this source.
By computer aided proof we show that at least x^{0.84} of the integers below x contain 1 in their forward orbit under the 3x+1 map.
Abstract of the paper that proves the bound; the body states it as π1(x) > x^0.84 for all sufficiently large x, and more generally for the count of integers whose orbit contains any fixed a not divisible by 3.
The source's own evidence bears what it asserts. The primary source. The paper establishes that feasibility of a linear program associated to Krasikov's difference inequalities yields exponential lower bounds for their positive monotone solutions, then reports a computation for the modulus 3^11 system giving the exponent 0.84; the bound holds for all sufficiently large x, and more generally for integers whose orbit contains any fixed value not divisible by 3. The result was refereed and published in Acta Arithmetica in 2003. Worth reading closely: It is the only proof of the claim; sections 2 and 6 give the main theorem and the computation that yields the exponent.
The Krasikov-Lagarias paper shows that at least N^0.84 of the positive numbers < N fall into the 4-2-1 cycle of the 3x+1 problem.
Comment on the OEIS entry for the Collatz map, summarizing the state of knowledge on the 3x+1 problem and citing Krasikov–Lagarias as the best partial result.
Asserted without evidence of the source's own. A one-sentence restatement of the Krasikov–Lagarias theorem in an encyclopedic sequence entry, offered without argument; only the search excerpt of the entry was read.
How these sources relate
- https://terrytao.wordpress.com/2019/09/10/almost-all-collatz-orbits-attain-almost-bounded-values/ restates https://arxiv.org/abs/math/0205002, faithfully. Tao restates the theorem exactly as the paper states it (a lower bound of order x^0.84 on the count of N ≤ x reaching 1, for all sufficiently large x), adding no evidence of his own and dropping no qualification.
- A006370 - OEIS (the Collatz or 3x+1 map) restates https://arxiv.org/abs/math/0205002, faithfully. The OEIS comment restates the theorem, credited to the paper, with no evidence of its own; "fall into the 4-2-1 cycle" is equivalent to the orbit reaching 1. The comment omits the "sufficiently large N" qualifier, a minor simplification that does not change the proposition.
Cite this claim: a formal citation with its evidence attached
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Created by extractor · Sep 13, 2026. Every judgment on this page is accompanied by a reasoning trace.