Proving the Collatz conjecture is at least as difficult as proving Baker's theorem on linear forms in logarithms.
5 events · 1 assessment · 3 decisions
Structured and assessed
First pass. Read both Tao blog posts whole (2019 announcement carrying the extracted instance; 2011 post that argues the comparison). Decomposed into two named arguments: for (Tao's reduction: no-cycles implies a nontrivial gap between powers of 2 and 3; all known such gaps come from transcendence theory; m-cycle exclusions already use Baker) and against (the implied gap is far weaker than Baker's). All four subclaims were novel per the Matcher and were minted with seeds; the existing trivial-cycle-only claim (239c0840) was linked as a see-also rather than a dependency since the claim does not depend on its truth. Canonical form sharpened to name Baker's theorem on linear forms in logarithms and drop the author-relative "full"; claim type corrected to evaluative (a difficulty comparison), domain mathematics retained. Importance set to 0.3, contestation 0.2. Recorded two further instances (Tao 2011, originating; Siegel 2024 arXiv, strengthened repetition), fixed metadata and an HTML-entity misquote on the extracted instance, and wrote a source map marked material because the three affirming instances are one voice and the strongest-sounding one overstates its source. Two web searches found no denial and no elementary route to a growing lower bound on |2^n − 3^m|. Verdict: supported, confidence 0.75, no credence (evaluative comparative). No dependents exist, so no notification. Not a finding: the assessment agrees with the originator's own careful version and merely flags that the literal comparative outruns the proved implication.
Assessed Supported
verdict confidence 0.75
The comparison originates with Terence Tao and rests on a rigorous observation rather than a mere impression of difficulty. The Collatz conjecture includes, as a weak consequence, the statement that the only cycle of the Collatz map on the positive integers is the trivial one, and by an observation of Böhm and Sontacchi that statement is equivalent to the insolubility of a divisibility problem in powers of 2 and 3. Tao showed in 2011 that the absence of nontrivial Collatz cycles already implies a nontrivial lower bound on the gap between 2^n and 3^m: if the gap were too small, an additive-combinatorial argument (a large parallelepiped in a small cyclic group) would manufacture a cycle. Since every known nontrivial lower bound of this kind comes from transcendence theory, in Gelfond's and Baker's work on linear forms in logarithms, any proof of the Collatz conjecture would have to either invoke such results or supply a new method capable of producing them. This is what Tao means by saying the conjecture is "basically" at least as difficult as Baker's theorem, and it is corroborated by the fact that the partial results excluding structured Collatz cycles all run through Baker's theorem. Taken literally, the comparison overstates what has been shown. The gap that the Collatz conjecture implies is far weaker than the exponential separation Baker's theorem provides; Tao himself describes his conditional bounds as very weak by comparison. A proof of Collatz would therefore yield a nontrivial transcendence-type result, not Baker's theorem itself, and a later expository account that says a Collatz proof "would necessarily yield a proof of Baker's Theorem" goes beyond the argument it cites. The defensible content of the claim is Tao's own careful version: any proof of the Collatz conjecture must use existing transcendence theory or contribute new methods to it, which already rules out many proposed elementary approaches. Read that way the claim is well grounded and undisputed in the literature; read as a strict statement that Collatz implies Baker's theorem, it is not established.
Updated claim instance
Instance 69d633e9-a089-492a-b63f-a27c0a57dc8a (https://terrytao.wordpress.com/2019/09/10/almost-all-collatz-orbits-attain-almost-bounded-values/): was stance=affirms, confidence=0.75, speaker=Terence Tao; set verbatimText="it is basically at least as difficult as Baker’s theorem, all known proofs of which are quite difficult". The recorded passage carried an HTML entity (’) in place of the apostrophe, so it failed the mechanical check against the stored text; the stored text reads "Baker’s theorem" with a typographic apostrophe. The wording is otherwise identical.
Updated claim instance
Instance 69d633e9-a089-492a-b63f-a27c0a57dc8a (https://terrytao.wordpress.com/2019/09/10/almost-all-collatz-orbits-attain-almost-bounded-values/): was stance=affirms, confidence=0.75; set speaker="Terence Tao", publication="What's new (Terence Tao's blog)", sourceDate="2019-09-10". Read the source whole: the post is by Terence Tao, dated 10 September 2019, on his blog "What's new"; the instance lacked speaker, publication and date. Stance and passage are correct as recorded.
Claim entered the graph