Every positive integer's Collatz orbit eventually reaches 1.
Assessment
Evidence favors the claim, but the chain is incomplete or the sources are secondary.
The Collatz conjecture (also called the 3n+1 problem) asserts that iterating the rule "halve an even number, triple an odd number and add one" from any positive integer eventually reaches 1. It is an open problem: no proof or counterexample is known, and no claimed proof has been accepted by the mathematical community despite a steady stream of attempts. It is nonetheless supported by evidence that mathematicians count as substantial, and it is widely believed to be true.
The evidence is of three kinds. First, exhaustive computation: every starting value up to 2^71 has been checked to reach 1, and this range, combined with cycle-length arguments, means any nontrivial cycle would have to contain more than roughly a hundred billion elements. Second, a probabilistic heuristic: on average each odd step followed by its halvings multiplies a number by about 3/4, so a typical orbit should drift downward. Third, rigorous almost-all theorems that partially confirm the heuristic: Terras showed that almost all orbits drop below their starting value, Krasikov and Lagarias showed that at least of order x^0.84 integers up to x reach 1, and Tao (2019) showed that almost all orbits, in logarithmic density, fall below any function tending to infinity.
None of this settles the question, because the conjecture requires two universal statements that almost-all results cannot deliver: that the only cycle is the trivial one through 1, 4, 2 and that no orbit is unbounded. A single exceptional integer would refute the conjecture while leaving every density theorem intact. Credible skepticism, voiced for instance by Alex Kontorovich, points out that the same heuristic predicts divergence for the closely related 5n+1 map, that Conway showed generalized Collatz-type problems to be algorithmically undecidable, and that the verified range is a vanishing fraction of the integers. These considerations do not amount to evidence that the conjecture is false; they explain why the evidence for it, though strong, is not decisive. A proof, a counterexample, or a demonstration of independence from standard axioms would resolve it. The probability that the conjecture is true is judged at about 0.88.
Full reasoning: the evidence and decisions behind this verdict
Status. The mathematics domain treats an open conjecture with evidence mathematicians count as supported, and an open conjecture with nothing beyond plausibility as unsupported. Collatz falls in the first category: the computational record, the random-walk heuristic, and the almost-all theorems (Terras, Korec, Krasikov and Lagarias, Tao) are all cited by the professional literature as evidence. Contested was considered and rejected: no credible mathematician asserts the negation. Kontorovich's 2019 remarks (reported at www.johndcook.com/blog/2019/09/16/collatz-conjecture-skepticism/) argue that the conjecture might be false, which is doubt about the evidence rather than a denial, and Tao's own post (terrytao.wordpress.com/2019/09/10/almost-all-collatz-orbits-attain-almost-bounded-values/) states the conjecture as open and says a full proof is out of reach.
Instances. The only recorded source is Tao's blog post, which poses the conjecture rather than asserting it; the two rows from that post were one passage recorded twice and have been read accordingly. The instance set therefore carries no stance signal, and the verdict rests on the evidence itself.
Evidence weighed. Barina's project reports verification of all starting values below 2^71 (2025; earlier 2^68 in the Journal of Supercomputing, 2021; code public at github.com/xbarin02/collatz). Hercher (Journal of Integer Sequences, 2023, cs.uwaterloo.ca/journals/JIS/VOL26/Hercher/hercher5.html) shows there are no Collatz m-cycles with m at most 91 and that a nontrivial cycle must have more than about 7.2 times 10^10 odd members. Tao's theorem gives, for any function tending to infinity, that almost all integers in logarithmic density have Collatz minimum below it; the post explains why the method stops at almost-all (the exceptional set of a local-in-time estimate can absorb the whole orbit distribution), which is why the boundedness half remains genuinely open rather than nearly done.
Material subclaims. The two required halves, no nontrivial cycle and no unbounded trajectory, are both open; the parent cannot be verified while either is. The cycle half is the better constrained (enormous cycle-length lower bounds plus the exact balance a cycle would require), the boundedness half is where informed doubt concentrates. The supporting density theorems are established results and are treated as reliable; their weight is limited by their almost-all form, not by any doubt about their proofs.
Credence. 0.88 reflects the strength of the heuristic and the computational record against three deductions: the 5n+1 analogy, where the same heuristic predicts divergence and nontrivial cycles are known; the history of heuristically plausible conjectures that failed at very large values (Mertens, Polya); and Conway's undecidability theorem for the generalized family, which shows no uniform method can exist and leaves open that the 3n+1 case could be true but unprovable in standard systems. Verdict confidence of 0.88 rather than higher because the line between supported and unsupported for open conjectures is a judgment about what counts as evidence, and a reader who discounts almost-all results and finite computation entirely would place it on the other side.
What would change the verdict. A machine-checked or refereed proof would make it verified; a counterexample or a proof of a divergent orbit or nontrivial cycle would make it contradicted; a credible mathematician publicly asserting the negation with an argument would make it contested. A claimed proof published in 2025 in Research in Mathematics (Taylor and Francis, "Unfolding the Collatz Tree") has not been accepted by the field and does not move the status; if it gains independent expert endorsement it would need to be recorded as its own claim about the proof.
Provenance. The support rests on primary results (Barina's computations, Hercher's cycle bounds, Tao's theorem) read directly or through their authors' own statements; the single recorded instance is Tao's post, which was read whole. The recorded passage of that instance was not found verbatim in the stored text because the stored copy strips mathematical notation; the sentence is present in the post as "Conjecture 1 (Collatz conjecture)" followed by the statement that the Collatz minimum equals 1 for all positive integers.
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.
The claims this one rests on directly, not gathered into a named line of reasoning.
- requiresa load-bearing premise: the parent is false without itsteward instructions →The only cycle of the Collatz map on the positive integers is the trivial cycle 1, 4, 2. ↗︎
- requiresa load-bearing premise: the parent is false without itsteward instructions →No positive integer has an unbounded Collatz trajectory. ↗︎
Because every starting value up to 2^71 has been checked to reach 1, no counterexample exists below that bound, and since any nontrivial cycle would have to contain an integer larger than every verified value, cycle-length arguments (Steiner, Simons and de Weger, Hercher) force such a cycle to have more than about 10^11 elements; the absence of counterexamples across this range supports the conjecture.
Granting that every starting value up to 2^71 reaches 1, the inference to the full conjecture is inductive: the verified range is a vanishing fraction of the integers, and heuristically plausible conjectures have failed at values far beyond any computation. The argument's firmest contribution is indirect, since the verified range feeds the cycle-length bounds that make a nontrivial cycle extremely constrained; it says little about a possible divergent orbit, which is where informed doubt concentrates.
Because almost all orbits drop below their starting value, because almost all orbits, in logarithmic density, fall below any function tending to infinity, and because at least of order x^0.84 integers up to x reach 1, the downward drift predicted by the random-walk heuristic (each Syracuse step multiplies by about 3/4 on average) is rigorously confirmed for typical integers, which supports the conjecture that every integer descends to 1.
The premises are established theorems, so the argument's weight lies entirely in the inferential step from "almost all" to "all", and that step is inductive rather than deductive: each result, including Tao's logarithmic-density theorem, is compatible with a density-zero set of exceptional orbits, and Tao's own account explains why the method cannot reach every integer. The argument therefore raises confidence in the conjecture without being able to establish it, and it gives no special protection against a single sporadic counterexample of the kind the 5n+1 analogy warns of.
Provenance
Where this claim has been said, linked to its canonical form.
One has {\mathrm{Col}_{\min}(N)=1} for all {N \in {\bf N}+1}.
Conjecture 1 (Collatz conjecture)
Asserted without evidence of the source's own. Tao states the conjecture as an open problem and says a full proof is out of reach; the post's own contribution is a theorem about almost all orbits, which he presents as evidence-adjacent progress rather than as settling the conjecture. Worth reading closely: It is the clearest short account of the strongest partial result toward the conjecture and of why the almost-all methods do not extend to every integer. The quoted passage was not found in the stored copy of this source.
Conjecture 1 (Collatz conjecture) One has
Statement of the Collatz (3x+1) conjecture: Col_min(N)=1 for all N in N+1, where Col_min is the minimal element of the Collatz orbit of N.
Asserted without evidence of the source's own.
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.