Schneider's eighth problem: at least one of e^e and e^{e²} is transcendental

Proved

Transcendence.exp_e_or_exp_e_sq_transcendental

by carlok · Sep 30, 2026 · Mathlib 0df444a (Lean v4.33.1)

diaz-modulus-leannumber-theorytranscendence

At least one of the two numbers

e^{e} \qquad\text{and}\qquad e^{e^{2}}

is transcendental.

This is the last of the eight open problems in Schneider's book (1957), as quoted by Waldschmidt (1973), p. 191.

Proof. Apply DiazModulus.two_algebraically_independent_of_exp_column at $x = y = (1, e)$ . Both pairs are linearly independent over $\mathbb{Q}$ because $e$ is irrational. The column $y_2 = e$ carries exactly $e^{e}$ and $e^{e^{2}}$ . If both were algebraic, all eight numbers would be algebraic over $\mathbb{Q}(e)$ , a field of transcendence degree one, against the theorem.

Novelty. None: this is the case $r = 1$ of the Solution du problème de Schneider in Waldschmidt (1973), p. 192, which the paper deduces from its Corollaire 1 (for rational $r$ , one of $e^{e^r}$ , $e^{e^{2r}}$ is transcendental). The substitution used here is Corollaire 1's, with $x_1$ and $x_2$ exchanged. By the note added in proof (p. 202), Brownawell solved the problem independently. The contribution of this node is the formal proof.

Preamble

import Mathlib

Formal statement

namespace Transcendence

/-- One of `e^e` and `e^{e²}` is transcendental.

This is the problem in the title of M. Waldschmidt, *Solution du huitième problème de Schneider*,
J. Number Theory 5 (1973), 191–202. It is the Théorème there
(`DiazModulus.two_algebraically_independent_of_exp_column`) at `x = y = (1, e)`: the column is
`e^e`, `e^{e²}`, and if both were algebraic, the eight numbers `1, e, 1, e, e, e^e, e^e, e^{e²}`
would all be algebraic over `ℚ(e)`, so no two of them could be algebraically independent. -/
theorem exp_e_or_exp_e_sq_transcendental :
    Transcendental ℚ (Complex.exp (Complex.exp 1)) ∨
      Transcendental ℚ (Complex.exp (Complex.exp 1 ^ 2)) := by
  sorry

end Transcendence

Source

The problem: T. Schneider, Einführung in die transzendenten Zahlen, Springer, Berlin, 1957, as quoted in M. Waldschmidt, Solution du huitième problème de Schneider, J. Number Theory 5 (1973), 191–202, p. 191. The solution: the same paper, p. 192 (Solution du problème de Schneider, r = 1, deduced from Corollaire 1); found independently by W. D. Brownawell, The algebraic independence of certain numbers related by the exponential function, J. Number Theory 6 (1974), 22–31. Formal proof: Diaz modulus mission, 30 September 2026 (C. Perassi).

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