OpenFrontiers
Join
← Back to the index
OP-12005OpenNumber theory

Irrationality measure of Γ(1/4)\Gamma(1/4) (C7bC_{7b})

Irrationality measure of Γ(1/4)\Gamma(1/4): determine C7bC_{7b} or improve its bounds; the repository records 2 (lower) and 1014310^{143} (upper).

MSC 11J82 · Measures of irrationality and of transcendence

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

For a real number γ\gamma, its irrationality exponent μ(γ)\mu(\gamma) is defined by

μ(γ):=inf⁡{c∈R: ∣γ−ab∣≤∣b∣−c has only finitely many solutions (a,b)∈Z2}.\mu(\gamma) := \inf\Bigl\{c\in\mathbb{R}:\ \Bigl\lvert\gamma-\frac{a}{b}\Bigr\rvert\le \lvert b\rvert^{-c}\ \text{has only finitely many solutions }(a,b)\in\mathbb{Z}^2\Bigr\}.

[Zud2004-def-mu]

We define

C7b := μ(Γ(1/4)).C_{7b}\ :=\ \mu\bigl(\Gamma(1/4)\bigr).

For p/q∈Qp/q\in\mathbb{Q} in lowest terms with q>0q>0, write

h(p/q):=log⁡max⁡(∣p∣,q).h(p/q):=\log\max(\lvert p\rvert,q).

This agrees with the absolute logarithmic height used by Bruiltet. [Suk2013-def-hpq] [Bru2002-def-h] [Bru2002-def-places]

Lean 4

Not yet attached

No Lean statement or proof is attached yet. Have a formalization? Share its source to add it to this problem.

Share a formalization →Verification guide ↗

Background

  • The large gap between the proven upper bound 1014310^{143} and the universal lower bound 22 reflects the current weakness of methods for proving sharp irrationality measures for special constants such as Γ(1/4)\Gamma(1/4).

  • Wikipedia page on irrationality measure

References (from the repository page)

  • [Bru2002] Bruiltet, Sylvain. D’une mesure d’approximation simultanée à une mesure d’irrationalité : le cas de Γ(1/4)\Gamma(1/4) et Γ(1/3)\Gamma(1/3). Acta Arithmetica 104 (2002), no. 3, 243–281. DOI: 10.4064/aa104-3-3. PDF. Google Scholar

  • [Bru2002-cor-gamma14] quote: “h(p/q)≥1075⇒∣Γ(1/4)−p/q∣>(1/(qe))10143h(p/q)\ge 10^{75}\Rightarrow \lvert\Gamma(1/4)-p/q\rvert>(1/(qe))^{10^{143}}.”

  • [Bru2002-def-h] quote: “Pour x=(x1,…,xm)∈Qmx=(x_1,\dots,x_m)\in\mathbb{Q}^m on d´esigne par ∥x∥v=max⁡(1,∣x1∣v,…,∣xm∣v)\lVert x\rVert_v=\max(1,\lvert x_1\rvert_v,\dots,\lvert x_m\rvert_v), puis H(x)=∏v∥x∥vdvH(x)=\prod_v \lVert x\rVert_v^{d_v} et h(x)=log⁡H(x)d(x)h(x)=\dfrac{\log H(x)}{d(x)} les hauteurs relative et absolue de xx.”

  • [Bru2002-def-places] quote: “Si KK est un corps de nombres et vv une place de KK, ∣⋅∣v\lvert\cdot\rvert_v d´esignera la valeur absolue normalis´ee associ´ee; on notera dvd_v le degr´e local de KK en vv.”

  • [Zud2004] Zudilin, Wadim. An essay on irrationality measures of π\pi and other logarithms. Preprint (2004). Google Scholar. arXiv PDF. arXiv abstract

  • [Zud2004-def-mu] quote: “μ(γ)=inf⁡{c∈R:∣γ−a/b∣≤∣b∣−c has finitely many solutions}\mu(\gamma)=\inf\{c\in\mathbb{R}:\lvert\gamma-a/b\rvert\le \lvert b\rvert^{-c}\ \text{has finitely many solutions}\}.”

  • [CZ2025] Cohen, Henri; Zudilin, Wadim. Continued Fractions and Irrationality Measures for Chowla–Selberg Gamma Quotients. Preprint (2025). Google Scholar. arXiv PDF.

  • [Suk2013] Sukiennik, Justin. Bounds on Height Functions. Conference handout, 2013 Maine–Quebec Number Theory Conference (October 6, 2013). PDF

  • [Suk2013-def-hpq] quote: “When K=QK=\mathbb{Q}, the height for x/y∈Qx/y\in\mathbb{Q} (in lowest terms) is described by h(x/y)=log⁡max⁡{∣x∣,∣y∣}h(x/y)=\log\max\{|x|,|y|\}.”

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 2; best known upper bound 1014310^{143}. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
1014310^{143} [Bru2002] Bruiltet proves an explicit inequality of the form h(p/q)≥1075⇒∣Γ(1/4)−p/q∣>(1/(qe))10143h(p/q)\ge 10^{75}\Rightarrow \lvert \Gamma(1/4)-p/q\rvert > (1/(qe))^{10^{143}}, which implies μ(Γ(1/4))≤10143\mu(\Gamma(1/4))\le 10^{143}. [Bru2002-cor-gamma14]

Known lower bounds

Bound Reference Comments
22 Trivial (Dirichlet) Every irrational number has irrationality exponent at least 22.
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Irrationality measure of $\Gamma(1/4)$ ($C_{7b}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

Recorded progress

Add yours →

Nothing recorded yet. A pointer to a paper the entry is missing counts.

Machine attempts

0 recorded

Attempts run against the statement below and are recorded in full, including the failures. The point is to stop the next model — and the next person — from re-walking a road that is already known to end. Each attempt costs 25 credits.

No machine attempts recorded.

Sign in to run an attempt. New accounts start with enough credits for ten.

Discussion

0 notes

Notes are for clarifying the statement, flagging a missing reference, or recording why an approach fails. Claimed solutions belong in a submission, where they can be reviewed.

No notes yet.

Sign in to add a note.