Open problem, Arithmetic and number theory, posed 1882

Irrationality of e + π and e·π

Open

Are the real numbers $e + \pi$ and $e\pi$ irrational? More generally, are $e$ and $\pi$ algebraically independent over $\mathbb{Q}$ ?

Research frontier as of 2026

As of 2026, it remains unknown whether $e + \pi$ or $e\pi$ is irrational. While Nesterenko (1996) proved the algebraic independence of $\pi$ and $e^\pi$ by exploiting the differential equations satisfied by Eisenstein series at the CM point $\tau = i$ , the number $e = e^1$ corresponds to evaluating the exponential function at $1$ rather than at a rational multiple of $\pi i$ , so modular-function methods do not simultaneously capture $e$ and $\pi$ . Meanwhile, the Lindemann–Weierstrass and Baker theorems handle values of $e^z$ at algebraic points $z$ , whereas $i\pi$ is transcendental.

Best known results

Elementary symmetric polynomial relation: because $e$ and $\pi$ are roots of $x^2 - (e+\pi)x + e\pi = 0$ and are transcendental, at least one of $e + \pi$ and $e\pi$ is transcendental.
Nesterenko (1996): $\pi$ and $e^\pi$ are algebraically independent over $\mathbb{Q}$ (so $\pi + e^\pi$ and $\pi e^\pi$ are transcendental).

Tools and where they stop

Tool	Achieved	Where it stops
Lindemann–Weierstrass theorem and linear forms in logarithms	Proves transcendence of $e$ ( $e^1$ is transcendental) and of $\pi$ ( $e^{i\pi} = -1$ is algebraic, so $i\pi$ is transcendental)	Requires either algebraic inputs $z$ or algebraic values $e^z$ ; cannot relate $z_1 = 1$ (where $e^{z_1} = e$ is transcendental) simultaneously with $z_2 = i\pi$ (where $z_2$ itself is transcendental)
Zero estimates for Ramanujan / Eisenstein modular functions (Nesterenko's method)	Proves that $\pi$ , $e^\pi$ , and $\Gamma(1/4)$ are algebraically independent over $\mathbb{Q}$	Values of the modular nome $q = e^{2\pi i \tau}$ at CM points $\tau$ produce numbers like $e^{-\pi}$ , not $e = e^1$

Open questions

Are $e + \pi$ and $e\pi$ both irrational (and transcendental)?
Are $e$ and $\pi$ algebraically independent over $\mathbb{Q}$ ?

References

Alan Baker (1975). Transcendental Number Theory · DOI:10.1017/CBO9780511565977
Yuri V. Nesterenko (1996). Modular functions and transcendence questions · DOI:10.1070/SM1996v187n09ABEH000158
Michel Waldschmidt (2000). Diophantine Approximations and Transcendental Numbers · DOI:10.1007/978-3-662-11569-5