The strong five exponentials conjecture implies that e^{λμ} is transcendental for non-zero logarithms λ, μ, and e^{π²} in particular

Here $\mathcal{L}$ is the set of logarithms of algebraic numbers, and $\widetilde{\mathcal{L}}$ is the $\overline{\mathbb{Q}}$-vector space spanned by $1$ and $\mathcal{L}$. Waldschmidt's strong five exponentials conjecture is carried as the hypothesis hS5, in the form of DiazModulus.diaz_of_strong_five_exponentials: if $x_1, x_2$ are linearly independent over $\mathbb{Q}$ and so are $y_1, y_2$, if $\eta \neq 0$, $\alpha_{ij}$ and $\beta$ are algebraic, and if the five numbers $e^{x_iy_j - \alpha_{ij}}$ ($i, j = 1, 2$) and $e^{\eta x_2/x_1 - \beta}$ are algebraic, then $x_iy_j = \alpha_{ij}$ for all $i, j$ and $\eta x_2 = \beta x_1$.

Then $e^{\lambda\mu}$ is transcendental for all non-zero $\lambda, \mu \in \mathcal{L}$: $(\log a)(\log b) \neq \log c$ for algebraic $a, b, c$ and non-zero logarithms $\log a$, $\log b$. In particular $e^{\lambda^2}$ is transcendental for every non-zero $\lambda \in \mathcal{L}$, and so is $e^{\pi^2}$.

Unconditionally, it is known only that $e^{\pi^2}$ is transcendental or $e$ and $\pi$ are algebraically independent (Brownawell and Waldschmidt; on the board as DiazModulus.algebraicIndependent_e_pi_of_exp_pi_sq_algebraic).

Proof. By Hermite–Lindemann a non-zero logarithm of an algebraic number is irrational, so $(1, \lambda)$ and $(1 + \mu, \mu)$ are linearly independent over $\mathbb{Q}$. If $e^{\lambda\mu}$ were algebraic, the conjecture would apply to $x = (1, \lambda)$, $y = (1 + \mu, \mu)$, $\eta = 1$, $\alpha_{11} = 1$ and $\alpha_{12} = \alpha_{21} = \alpha_{22} = \beta = 0$: the five numbers are $e^{\mu}$, $e^{\mu}$, $e^{\lambda}e^{\lambda\mu}$, $e^{\lambda\mu}$ and $e^{\lambda}$. Its conclusion $x_2y_2 = \alpha_{22}$ reads $\lambda\mu = 0$. For $\pi$, take $\lambda = \mu = i\pi$, a logarithm of $-1$, and note that $e^{-\pi^2} = 1/e^{\pi^2}$.

Novelty. None: the implication and the choice of parameters are Waldschmidt's (1988, p. 379). His 2005 survey derives the cases $e^{\lambda^2}$ and $e^{\pi^2}$, with $x = y = (1, \lambda)$ (pp. 4–5 of the author's version). The contribution of this node is the formal proof.