six exponentials theorem

Complex numbers $x_1,x_2,\mathrm{\dots },x_n$ are $\mathbb{Q}$-linearly independent if the only rational numbers$r_1,r_2,\mathrm{\dots },r_n$ with

$$r_1{x}_1+r_2{x}_2+\mathrm{\cdots }+r_n{x}_n=0$$

are $r_1=r_2=\mathrm{\cdots }=r_n=0$.

Six Exponentials Theorem:If $x_{\mathrm{1}}\mathrm{,}{x}_{\mathrm{2}}\mathrm{,}{x}_{\mathrm{3}}$ are $\mathrm{Q}$-linearly independent, and $y_{\mathrm{1}}\mathrm{,}{y}_{\mathrm{2}}$ are also $\mathrm{Q}$-linearly independent, then atleast one of the six numbers $\exp \mathit{}\mathrm{(}{x}_i\mathit{}{y}_j\mathrm{)}$ is transcendental.

This is weaker than the Four Exponentials Conjecture.

Four Exponentials Conjecture:Given four complex numbers $x_{\mathrm{1}}\mathrm{,}{x}_{\mathrm{2}}\mathrm{,}{y}_{\mathrm{1}}\mathrm{,}{y}_{\mathrm{2}}$, either $x_{\mathrm{1}}\mathrm{/}{x}_{\mathrm{2}}$ or $y_{\mathrm{1}}\mathrm{/}{y}_{\mathrm{2}}$ is rational, or one of the fournumbers $\exp \mathit{}\mathrm{(}{x}_i\mathit{}{y}_j\mathrm{)}$ is transcendental.

For the history of the six exponentials theorem, we quote briefly from [6, p. 15]:

The six exponentials theorem occurs for the first time in a paper by L. Alaoglu and P. Erdős[1], when these authors try to prove Ramanujan’s assertion that the quotient of two consecutivesuperior highly composite numbers is a prime, they need to know that if $x$ is a real number such that$p_1^x$ and $p_2^x$ are both rational numbers, with $p_1$ and $p_2$ distinct prime numbers, then $x$ is aninteger. However, this statement (special case of the four exponentials conjecture) is yet unproven. Theyquote C. L. Siegel and claim that $x$ indeed is an integer if one assumes $p_i^x$ to be rational for three distinct primes $p_i$. This is just a special case of the six exponentials theorem. They deduce thatthe quotient of two consecutive superior highly composite numbers is either a prime, or else a product of twoprimes.
The six exponentials theorem can be deduced from a very general result of Th. Schneider [4]. Thefour exponentials conjecture is equivalent to the first of the eight problems at the end of Schneider’s book[5]. An explicit statement of the six exponentials theorem, together with a proof, has beenpublished independently and at about the same time by S. Lang [2, Chapter 2] and K. Ramachandra[3, Chapter 2]. They both formulated the four exponentials conjecture explicitly.