The irrationality measure of Γ(1/4)

For a real number γ, its irrationality exponent μ(γ) is defined by μ(γ) := infBiglc∈ℝ: Bigllvertγ-a/bBigrrvert≤ lvert brvert^-c has only finitely many solutions (a,b)∈ℤ^2Bigr. <a href="#Zud2004-def-mu">[Zud2004-def-mu]</a> We define C_7b := μbigl(Γ(1/4)bigr).

Current state (tracked automatically)

Best known lower bound for constant 7b: 2 (Trivial (Dirichlet), Every irrational number has irrationality exponent at least 2).

All tracked facts and sources

Fact	Value	Source
Best lower bound with an explicit value (constant 7b)	2 Trivial (Dirichlet) — Every irrational number has irrationality exponent at least 2.	Tao's optimization constants

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Tao's optimization constants (checked 2026-11-12). How it works.

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Cite

@misc{cairn-constant-7b-the-irrationality-measure-of-1-4,
  title        = {The irrationality measure of Γ(1/4)},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/constant-7b-the-irrationality-measure-of-1-4}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-13}
}

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

Description of constant

For a real number $γ$ , its irrationality exponent $μ (γ)$ is defined by $μ (γ) := in f {c \in R : γ - \frac{a}{b} \leq ∣ b ∣^{- c} has only finitely many solutions (a, b) \in Z^{2}} .$

We define

$C_{7 b} := μ (Γ (1/4)) .$

For $p / q \in Q$ in lowest terms with $q > 0$ , write

$h (p / q) := lo g max (∣ p ∣, q) .$

This agrees with the absolute logarithmic height used by Bruiltet. <a href="#Suk2013-def-hpq">[Suk2013-def-hpq]</a> <a href="#Bru2002-def-h">[Bru2002-def-h]</a> <a href="#Bru2002-def-places">[Bru2002-def-places]</a>

Known upper bounds

Bound	Reference	Comments
$1 0^{143}$	<a href="#Bru2002">[Bru2002]</a>	Bruiltet proves an explicit inequality of the form $h (p / q) \geq 1 0^{75} \Rightarrow ∣ Γ (1/4) - p / q ∣ > (1/ (q e))^{1 0^{143}}$ , which implies $μ (Γ (1/4)) \leq 1 0^{143}$ . <a href="#Bru2002-cor-gamma14">[Bru2002-cor-gamma14]</a>

Known lower bounds

Bound	Reference	Comments
$2$	Trivial (Dirichlet)	Every irrational number has irrationality exponent at least $2$ .

Additional comments and links

The large gap between the proven upper bound $1 0^{143}$ and the universal lower bound $2$ reflects the current weakness of methods for proving sharp irrationality measures for special constants such as $Γ (1/4)$ .

Wikipedia page on irrationality measure

References

<a id="Bru2002"></a>[Bru2002] Bruiltet, Sylvain. D’une mesure d’approximation simultanée à une mesure d’irrationalité : le cas de $Γ (1/4)$ et $Γ (1/3)$ . Acta Arithmetica 104 (2002), no. 3, 243–281. DOI: 10.4064/aa104-3-3. PDF. Google Scholar
<a id="Bru2002-cor-gamma14"></a>[Bru2002-cor-gamma14] quote: “ $h (p / q) \geq 1 0^{75} \Rightarrow ∣ Γ (1/4) - p / q ∣ > (1/ (q e))^{1 0^{143}}$ .”

<a id="Bru2002-def-h"></a>[Bru2002-def-h] quote: “Pour $x = (x_{1}, \dots, x_{m}) \in Q^{m}$ on d´esigne par $∥ x ∥_{v} = max (1, ∣ x_{1} ∣_{v}, \dots, ∣ x_{m} ∣_{v})$ , puis $H (x) = \prod_{v} ∥ x ∥_{v}^{d_{v}}$ et $h (x) = \frac{lo g H ( x )}{d ( x )}$ les hauteurs relative et absolue de $x$ .”

<a id="Bru2002-def-places"></a>[Bru2002-def-places] quote: “Si $K$ est un corps de nombres et $v$ une place de $K$ , $∣ \cdot ∣_{v}$ d´esignera la valeur absolue normalis´ee associ´ee; on notera $d_{v}$ le degr´e local de $K$ en $v$ .”

<a id="Zud2004"></a>[Zud2004] Zudilin, Wadim. An essay on irrationality measures of $π$ and other logarithms. Preprint (2004). Google Scholar. arXiv PDF. arXiv abstract
<a id="Zud2004-def-mu"></a>[Zud2004-def-mu] quote: “ $μ (γ) = in f c \in R : ∣ γ - a / b ∣ \leq ∣ b ∣^{- c} has finitely many solutions$ .”

<a id="CZ2025"></a>[CZ2025] Cohen, Henri; Zudilin, Wadim. Continued Fractions and Irrationality Measures for Chowla–Selberg Gamma Quotients. Preprint (2025). Google Scholar. arXiv PDF.

<a id="Suk2013"></a>[Suk2013] Sukiennik, Justin. Bounds on Height Functions. Conference handout, 2013 Maine–Quebec Number Theory Conference (October 6, 2013). PDF
<a id="Suk2013-def-hpq"></a>[Suk2013-def-hpq] quote: “When $K = Q$ , the height for $x / y \in Q$ (in lowest terms) is described by $h (x / y) = lo g max ∣ x ∣, ∣ y ∣$ .”

What counts as progress

A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
A formal proof (Lean) of a known bound, or a precise error in a claimed one.
New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 87b4ebbd426f, Apache License 2.0; reformatted for this page. New records should also be reported there.