If log 2 is algebraic over ℚ(π), then exp √((log 2)² + π²) is transcendental

The smallest open instance of Diaz's conjecture, seen from the homogeneous side.

Suppose that $\log 2$ is algebraic over $\mathbb{Q}[i\pi]$; equivalently, $\log 2$ and $\pi$ are algebraically dependent. Then

is transcendental.

The number $\sqrt{(\log 2)^{2}+\pi^{2}} = |\log 2 + i\pi|$ is the modulus of a logarithm of $-2$. Whether it is algebraic is the smallest open instance of Diaz's conjecture. This node says that it is not a logarithm of an algebraic number, provided $\log 2$ and $\pi$ are algebraically dependent. Whether they are is not known.

Proof. As in DiazModulus.exp_abs_transcendental_of_conj_algebraic, at $\lambda = \log 2 + i\pi$. Here $\lambda$ and $\bar\lambda = \log 2 - i\pi$ are both algebraic over $\mathbb{Q}[i\pi]$, so the entries of $\begin{pmatrix} |\lambda| & \bar\lambda \\ \lambda & |\lambda| \end{pmatrix}$ generate a field of transcendence degree one and DiazModulus.four_exponentials_trdeg_one applies.

Attribution. Known: this is the example that D. Roy and M. Waldschmidt give after their Corollaire 7.4, Ann. Sci. École Norm. Sup. (4) 30 (1997), p. 792. The contribution of this node is the formal proof.