Conjecture 3.4 — Strong Four Exponentials Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Write $\overline{\mathbb Q}$ for the algebraic numbers in $\mathbb C$ and set $\mathcal L=\{\lambda\in\mathbb C:e^\lambda\in\overline{\mathbb Q}^{\times}\}$. Define $\widetilde{\mathcal L}=\operatorname{span}_{\overline{\mathbb Q}}(\{1\}\cup\mathcal L)$: finite algebraic linear combinations of $1$ and arbitrary complex logarithms of nonzero algebraic numbers.

Statement recorded in the supplied source.

If $x_1,x_2$ are linearly independent over $\overline{\mathbb Q}$ and $y_1,y_2$ are also linearly independent over $\overline{\mathbb Q}$, must at least one $x_i y_j$ ($1\le i,j\le2$) lie outside $\widetilde{\mathcal L}$? [S2]

Short English statement

Can both pairs of directions, each linearly independent over the algebraic numbers, have all four products in the algebraic span of one and logarithms of algebraic numbers?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700022 (AMR-086-0022). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Conjecture 3.4 and preceding definition of the logarithmic vector space. https://arxiv.org/abs/math/0312440.