Zaremba’s conjecture constant

Description of constant

Zaremba’s conjecture concerns denominators of rational numbers $b/d\in(0,1)$ whose finite continued fraction expansions have all partial quotients bounded by an absolute constant. [Kan2021-conj-reformulation]

Equivalently (in the standard notation of the subject), the conjecture predicts that there exists an integer $A>1$ such that every positive integer appears among the denominators that admit a continued-fraction expansion with all partial quotients in ${1,2,\ldots,A}$. [BK2014-conj-1.1-A5]

We define

\[C_{58}\ :=\ A_{\mathrm{Zar}},\]

where $A_{\mathrm{Zar}}$ is the least integer $A$ (if it exists) such that every positive integer $d$ occurs as a denominator of some reduced fraction $b/d$ whose finite continued fraction has all partial quotients $\le A$. If no such $A$ exists, set $A_{\mathrm{Zar}}:=+\infty$.

Zaremba conjectured that $A=5$ suffices, and moreover the $A=4$ version is known to be false (explicit counterexamples $d=54$ and $d=150$ are known), giving the rigorous lower bound

\[A_{\mathrm{Zar}}\ \ge\ 5.\]

[Kan2021-A4-counterexamples]

The strongest proven result toward finiteness is that the conjecture holds for a density-one set of denominators: Bourgain–Kontorovich prove a density-one statement and record an explicit choice $A=50$ for it. [BK2014-density-one] [BK2014-A50-suffices]

Huang later improved the density-one statement to partial quotients bounded by $5$ (still for a density-one subset of denominators, not all integers). [Hua2015-density-one-A5]

Known upper bounds

No finite upper bound for $A_{\mathrm{Zar}}$ is currently known (the conjecture remains open). [Kan2021-unproved-50years]

Bound	Reference	Comments
$+\infty$		Trivial.

Known lower bounds

Bound	Reference	Comments
$5$	[Kan2021]	The $A=4$ version fails (counterexamples $d=54,150$), so any universal bound must satisfy $A_{\mathrm{Zar}}\ge 5$. [Kan2021-A4-counterexamples]

Additional comments and links

Conjectural value. Zaremba suggested that $A_{\mathrm{Zar}}=5$. [BK2014-conj-1.1-A5] [Kan2021-A4-counterexamples]
Applications. Zaremba’s conjecture has applications to numerical integration and pseudorandom number generation (via low-discrepancy constructions). [BK2014-applications]
Wikipedia page on Zaremba’s conjecture

References

[BK2014] Bourgain, Jean; Kontorovich, Alex. On Zaremba’s conjecture. Annals of Mathematics 180 (2014), no. 1, 137–196. DOI: https://doi.org/10.4007/annals.2014.180.1.3. arXiv PDF: https://arxiv.org/pdf/1107.3776.pdf. Google Scholar
- [BK2014-conj-1.1-A5] loc: Annals PDF p.2, Sec. 1.1 quote: “That is, the conjecture predicts the existence of some integer $A > 1$ so that $D{1,2,…,A} = N$. In fact, Zaremba suggested that $A = 5$ may already be sufficient.”
- [BK2014-applications] loc: Annals PDF p.2, Sec. 1.1 quote: “Zaremba’s conjecture has important applications to numerical integration and pseudorandom number generation”
- [BK2014-density-one] loc: Annals PDF p.1, Abstract quote: “We confirm this conjecture for a set of density one.”
- [BK2014-A50-suffices] loc: Annals PDF p.2, Sec. 1.1 quote: “In particular, $A = 50$ suffices.”
[Hua2015] Huang, Shinn-Yih. An Improvement to Zaremba’s Conjecture. Geometric and Functional Analysis 25 (2015), 860–914. DOI: https://doi.org/10.1007/s00039-015-0327-6. arXiv PDF: https://arxiv.org/pdf/1310.3772.pdf. Google Scholar
- [Hua2015-density-one-A5] loc: arXiv PDF p.1, Abstract quote: “We prove there exists a density one subset $D \subset \mathbb{N}$ such that each $n \in D$ is the denominator of a finite continued fraction with partial quotients bounded by $5$.”
[Kan2021] Kan, I. D. A strengthening of the Bourgain–Kontorovich method: three new theorems. Sbornik: Mathematics 212 (2021), no. 7, 921–964. DOI: https://doi.org/10.1070/SM9437. PDF: https://iopscience.iop.org/article/10.1070/SM9437/pdf. Google Scholar
- [Kan2021-unproved-50years] loc: Kan_2021_Sb._Math._212_921.pdf p.2, Sec. 1.1 quote: “The following unproved conjecture has challenged mathematicians for about 50 years.”
- [Kan2021-conj-reformulation] loc: Kan_2021_Sb._Math._212_921.pdf p.2, Sec. 1.1 quote: “That is, every $d \ge 1$ is represented as the denominator of a finite continued fraction $b/d$ with partial quotients bounded by $A$.”
- [Kan2021-A4-counterexamples] loc: Kan_2021_Sb._Math._212_921.pdf p.2, Sec. 1.1 quote: “In fact, Zaremba conjectured that $A = 5$ is sufficient in his conjecture. Why exactly $A = 5$? Because a similar conjecture for $A = 4$ fails to hold for at least two values of $d$, namely, $d = 54$ and $d = 150$ (no other counterexamples are known!).”

Contribution notes

Prepared with assistance from ChatGPT 5.2 Pro.