= Six exponentials theorem
The six exponentials theorem states that if $x_1,x_2$ are complex numbers linearly independent over $\mathbb Q$, and $y_1,y_2,y_3$ are also linearly independent over $\mathbb Q$, then at least one of the six numbers
$$
e^{x_i y_j},\qquad 1\leq i\leq2,\quad1\leq j\leq3,
$$
is a <transcendental number>. It does not require the inputs themselves to be algebraic. Both independence hypotheses are essential.<Linear Independence of Logarithms of Algebraic Numbers by Michel Waldschmidt>[Chapter1, Theorem1.7]{ref}
For an application, suppose a real $t$ makes $2^t,3^t,5^t$ all algebraic. If $t$ were irrational, choose $x_1=1,x_2=t$ and $y_1=\log2,y_2=\log3,y_3=\log5$. The logarithms are rationally independent: clearing denominators and exponentiating would otherwise give a nontrivial integer relation $2^a3^b5^c=1$, contradicting unique prime factorization. All six outputs would be algebraic, a contradiction. Hence $t$ must be rational.
With only two independent $y_j$, the analogous assertion is the <four exponentials conjecture>, not this theorem.
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