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At least one of π+e\pi + eπ+e, πe\pi eπe is transcendental

Proved
FCP.Transcendence.transcendental_pi_add_e_or_pi_mul_e

by Lucas · Sep 15, 2026 · Mathlib 0df444a (Lean v4.33.1)

analysisnumber-theory

A known warm-up. At least one of π+e\pi + eπ+e and πe\pi eπe is transcendental. Indeed, if both were algebraic, then π\piπ and eee would be roots of the quadratic X2−(π+e)X+πeX^2 - (\pi+e)X + \pi eX2−(π+e)X+πe over Q‾\overline{\mathbb{Q}}Q​ and hence algebraic, contradicting the Lindemann--Weierstrass theorem. This is the elementary anchor of the transcendence group and a realistic formalization target.

Preamble
import Mathlib

open Real
Formal statement
namespace FCP.Transcendence

theorem transcendental_pi_add_e_or_pi_mul_e :
    Transcendental ℚ (π + exp 1) ∨ Transcendental ℚ (π * exp 1) := by sorry

end FCP.Transcendence
Source
Formal Conjectures library (Google DeepMind), Apache-2.0, https://github.com/google-deepmind/formal-conjectures (FormalConjectures/Wikipedia/Transcendental.lean); https://en.wikipedia.org/wiki/Transcendental_number
Read-back

What the Lean code literally says, in plain math · Aristotle by Harmonic (non-blind: same agent that drafted the statements)

Non-blind read-back. This read-back was not written by an independent blind auditor: it was written by the same agent that drafted the Lean statement, with full knowledge of the intended meaning and of the source material. It is therefore not independent testimony and must not be mistaken for it; a reviewer who wants genuine blind testimony should commission it separately.

The disjunction: either the real number π+exp⁡(1)\pi + \exp(1)π+exp(1) is transcendental over Q\mathbb{Q}Q, or the real number π⋅exp⁡(1)\pi \cdot \exp(1)π⋅exp(1) is transcendental over Q\mathbb{Q}Q. Only one of the two disjuncts is asserted to hold; no claim is made about which.

Human review
  • Endorsed by Shuze Chen · Sep 17, 2026

  • Endorsed by Lucas · Sep 17, 2026

    Confirmed by the mission captain (proposal self-audit).

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