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T. Tao and the Syracuse conjecture

Jean-Paul AlloucheEuropean Mathematical Society MagazinePublished Feb 25, 2022doi 10.4171/mag/64 Original︎ ↗Archived copy

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For almost all N in natural density, the Collatz orbit of N drops below N.
Supports. Allouche states the theorem and gives the shape of its proof: the parity of the first a iterates depends only on the residue class modulo 2^a, the number of odd steps is distributed binomially, and a tail estimate for binomial sums finishes the argument. He also records that the result was found independently by Everett, Möller, Heppner and Terras, and remarks that the coincidence suggests it is not very difficult. Sep 17, 2026

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MAG 123 pp. 12–160 passages

T. Tao and the Syracuse conjecture1 passage

Jean-Paul Allouche0 passages

Sorbonne Université, Paris, France0 passages

Abstract0 passages

1 Introduction0 passages

2 First steps0 passages

3 Going a bit further0 passages

4 The result obtained by T. Tao1 passage

References0 passages

Cite this article2 passages

1 Introduction11 passages

2 First steps9 passages · 1 marked

Theorem .

For almost all N in natural density, the Collatz orbit of N drops below N.VerifiedReading · this claim’s Steward

Supports. Allouche states the theorem and gives the shape of its proof: the parity of the first a iterates depends only on the residue class modulo 2^a, the number of odd steps is distributed binomially, and a tail estimate for binomial sums finishes the argument. He also records that the result was found independently by Everett, Möller, Heppner and Terras, and remarks that the coincidence suggests it is not very difficult.

For all i in [ 0 , a ] , we have0 passages

{ j ∈ [ 0 , 2 a − 1 ] ∣ α ( a , j ) = i } = ( i a ) .4 passages

A proof of this lemma suggested by G. Tenenbaum uses the relation3 passages

{ n ≤ x ∣ ∃ k , f k ( n ) < n } = x + O ( x 1 − δ )3 passages

3 Going a bit further2 passages

{ n ≤ x ∣ ∃ k , f k ( n ) = 1 } > x d1 passage

4 The result obtained by T. Tao38 passages

References14 references

Cite this article1 passage

DOI 10.4171/MAG/642 passages

Back to Issue 1230 passages

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Kind: journal article
Authors: Jean-Paul Allouche
Publisher: European Mathematical Society Magazine
Published: Feb 25, 2022
DOI: 10.4171/mag/64
Link: https://euromathsoc.org/magazine/articles/64
Archived copy: https://web.archive.org/web/20261008184339/https://euromathsoc.org/magazine/articles/64
Stored text: 26,319 characters, retrieved Nov 11, 2026 · 53 sections, 93 passages
Claims extracted: Not recorded
Versions: One known
Copies: None recorded
Corrections: None recorded
Facts checked: Nov 11, 2026

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