Is e + π transcendental?

number-theory · transcendence · algebraic-numbers

Is the real number e + π transcendental over the rational numbers? Here e = exp(1), and π is the usual circle constant. Transcendental means that no nonzero polynomial with rational coefficients has e + π as a root. This is stronger than asking whether e + π is irrational.

Why it matters
Knowing that e and π are individually transcendental says surprisingly little about their sum. This concrete question exposes the gap between transcendence of individual constants and their algebraic independence.

One possible first attack
Prove the known weaker alternative: at least one of e + π and eπ is transcendental. If both were algebraic, e and π would be roots of X² − (e + π)X + eπ and hence algebraic over the algebraic numbers, contradicting their transcendence. This argument deliberately does not decide which quantity is transcendental.

Mathematical status
Open (checked 2026-09-11). Even the irrationality of e + π is not known in the cited references. Algebraic independence of e and π would imply the target, but that stronger statement is also conjectural; it must not be inserted as an axiom.

Formal availability
A reviewed proposition definition is supplied in this corpus. A target is a question to prove, not a proof. Use the deployed target and proof-environment records for the exact accepted declaration, namespace, source artifact, toolchain and current availability; local corpus provenance does not establish deployment.

Sources and provenance
Waldschmidt, Introduction to Transcendental Number Theory 8 (2021), slides 14 and 30: Schanuel and the open algebraic-independence question for e and π (checked 2026-09-11): https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/IntroductionTranscendentalNumbersPart8.pdf
Reference for the unresolved irrationality and transcendence of e + π (checked 2026-09-11): https://mathworld.wolfram.com/e.html
Definition and the known alternative that at least one of e + π and eπ is transcendental (checked 2026-09-11): https://en.wikipedia.org/wiki/Transcendental_number
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/Wikipedia/Transcendental.lean
Local target: Corpus.WikipediaTranscendental.exp_add_pi_transcendental
Source SHA-256: 4cebebda90deb10dbeb7b8fb33035196ba3114a6f4748d562be7a3df7ddaac61
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
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