Editing Collatz conjecture (section)

==Supporting arguments==
Although the conjecture has not been proven, most mathematicians{{Citation needed|date=April 2025}} who have looked into the problem think the conjecture is true because experimental evidence and heuristic arguments support it.

===Experimental evidence===
The conjecture has been checked by computer for all starting values up to 2<sup>71</sup> ≈ {{val|2.36e21}}. All values tested so far converge to 1.<ref name=Barina>{{cite journal | last = Barina | first = David | title = Improved verification limit for the convergence of the Collatz conjecture | journal = The Journal of Supercomputing | year = 2025 | volume = 81 | issue = 810 | pages = 1–14 | doi = 10.1007/s11227-025-07337-0 | s2cid = 220294340 |url=https://link.springer.com/content/pdf/10.1007/s11227-025-07337-0.pdf }}</ref>

This computer evidence is still not rigorous proof that the conjecture is true for all starting values, as [[counterexamples]] may be found when considering very large (or possibly immense) positive integers, as in the case of the disproven [[Pólya conjecture]] and [[Mertens conjecture]].

However, such verifications may have other implications. Certain constraints on any non-trivial cycle, such as [[lower bound]]s on the length of the cycle, can be proven based on the value of the lowest term in the cycle. Therefore, computer searches to rule out cycles that have a small lowest term can strengthen these constraints.<ref name="Garner (1981)"/><ref name="Eliahou (1993)"/><ref name="Simons & de Weger (2005)"/>

===A probabilistic heuristic===
If one considers only the ''odd'' numbers in the sequence generated by the Collatz process, then each odd number is on average {{sfrac|3|4}} of the previous one.{{refn|{{named ref|name=Lagarias (1985)}} section "[http://www.cecm.sfu.ca/organics/papers/lagarias/paper/html/node3.html A heuristic argument"].}} (More precisely, the geometric mean of the ratios of outcomes is {{sfrac|3|4}}.) This yields a heuristic argument that every Hailstone sequence should decrease in the long run, although this is not evidence against other cycles, only against divergence. The argument is not a proof because it assumes that Hailstone sequences are assembled from uncorrelated probabilistic events. (It does rigorously establish that the [[p-adic numbers|2-adic]] extension of the Collatz process has two division steps for every multiplication step for [[almost all]] 2-adic starting values.)

===Stopping times===
As proven by [[Riho Terras (mathematician)|Riho Terras]], almost every positive integer has a finite stopping time.<ref name="Terras (1976)"/> In other words, almost every Collatz sequence reaches a point that is strictly below its initial value. The proof is based on the distribution of [[#As a parity sequence|parity vectors]] and uses the [[central limit theorem]].

In 2019, [[Terence Tao]] improved this result by showing, using logarithmic [[Probability density function|density]], that [[almost all]] (in the sense of logarithmic density) Collatz orbits are descending below any given function of the starting point, provided that this function diverges to infinity, no matter how slowly. Responding to this work, ''[[Quanta Magazine]]'' wrote that Tao "came away with one of the most significant results on the Collatz conjecture in decades".<ref name="Tao"/><ref>{{Cite web |last=Hartnett |first=Kevin |date=December 11, 2019 |title=Mathematician Proves Huge Result on 'Dangerous' Problem |url=https://www.quantamagazine.org/mathematician-proves-huge-result-on-dangerous-problem-20191211/ |website=Quanta Magazine}}</ref>

===Lower bounds===
In a [[computer-aided proof]], Krasikov and Lagarias showed that the number of integers in the interval {{math|[1,''x'']}} that eventually reach 1 is at least equal to {{math|''x''<sup>0.84</sup>}} for all sufficiently large {{mvar|x}}.<ref>{{Cite journal
 | last1 = Krasikov | first1 = Ilia
 | last2 = Lagarias | first2 = Jeffrey C. | author-link2 = Jeffrey Lagarias
 | year = 2003
 | title = Bounds for the 3''x''&nbsp;+&nbsp;1 problem using difference inequalities
 | journal = Acta Arithmetica
 | url = https://www.impan.pl/download/pdf/aa109-3-4
 | doi = 10.4064/aa109-3-4
 | mr = 1980260
 | volume = 109
 | issue = 3
 | pages = 237–258| arxiv = math/0205002
 | bibcode = 2003AcAri.109..237K
 | s2cid = 18467460
 }}</ref>