Source labels openChecked July 26, 2026

Open questions on transcendence of numbers

$\pi^{\pi^{\pi^\pi}}$ is not an integer.

Mathematical statement

$\pi^{\pi^{\pi^\pi}}$ is not an integer.

This would follow from $\pi^{\pi^{\pi^\pi}}$ being transcendental, but this formulation is of interest in its own right, as it could in principle be proven by direct computation. Reference:* YouTube

Statement source: Wikipedia statement material

Statement terms: Source-specific

Source-specific terms. Therefore does not assert reuse rights beyond attributed display.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

pi_pow_pi_pow_pi_pow_pi_not_integer

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem pi_pow_pi_pow_pi_pow_pi_not_integer : ¬ ∃ (n : ℤ), π ^ π ^ π ^ π = n := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Pinned Formal Conjectures sourceformalization YouTubebackground Wikipediabackground