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The Collatz conjecture, Littlewood-Offord theory, and powers of 2 and 3

Author not recordedWhat's newPublished Aug 25, 2011Original︎ ↗Archived copy

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Proving the Collatz conjecture is at least as difficult as proving Baker's theorem on linear forms in logarithms.
Supports. This is the origin of the comparison. The post proves the weaker conditional bound in full and sketches a stronger one, and it is careful to say that both fall well short of Baker's theorem; the claim it actually argues for is that any proof of Collatz must use or replace transcendence theory, not that it would reprove Baker's theorem. Sep 16, 2026

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The Collatz conjecture, Littlewood-Offord theory, and powers of 2 and 311 passages

Conjecture 3 (Reformulated weak Collatz conjecture) There does not exist and integers8 passages

In particular, as is odd, . Using the recursion1 passage

Since , we have1 passage

Conversely, suppose that Conjecture 3 failed. Then we have , integers11 passages

The heuristic number of solutions overall is then expected to be4 passages

A brief computation shows that20 passages · 1 marked

This bound is very weak when compared against the unconditional bound (7) . However, I know of no way to get a nontrivial separation property between powers of and powers of other than via transcendence theory methods. Thus, this result strongly suggests that any proof of the Collatz conjecture must either use existing results in transcendence theory, or else must contribute a new method to give non-trivial results in transcendence theory. (This already rules out a lot of possible approaches to solve the Collatz conjecture.)

Proving the Collatz conjecture is at least as difficult as proving Baker's theorem on linear forms in logarithms.SupportedReading · this claim’s Steward

Supports. This is the origin of the comparison. The post proves the weaker conditional bound in full and sketches a stronger one, and it is careful to say that both fall well short of Baker's theorem; the claim it actually argues for is that any proof of Collatz must use or replace transcendence theory, not that it would reprove Baker's theorem.

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The Collatz conjecture, Littlewood-Offord theory, and powers of 2 and 3

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(p,q)-adic Analysis and the Collatz Conjectureunknown1 claimDiverges · strongerDraws its statement from it more strongly than this source supports, on 1 claim.

Almost all Collatz orbits attain almost bounded valuesunknownSep 10, 20191 claimDiverges · strongerRestates it more strongly than this source supports, on 1 claim.

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Kind: unknown
Authors: Not recorded
Publisher: What's new
Published: Aug 25, 2011
DOI: None
Link: https://terrytao.wordpress.com/2011/08/25/the-collatz-conjecture-littlewood-offord-theory-and-powers-of-2-and-3/
Archived copy: https://web.archive.org/web/20261011085408/https://terrytao.wordpress.com/2011/08/25/the-collatz-conjecture-littlewood-offord-theory-and-powers-of-2-and-3/
Stored text: 56,425 characters, retrieved Sep 16, 2026 · 226 sections, 554 passages
Claims extracted: Not recorded
Versions: One known
Copies: None recorded
Corrections: None recorded
Facts checked: Nov 9, 2026

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Nov 9, 2026DividedSegmenter · Divided into 779 parts by its own structure
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