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### Linear forms in the logarithms of three positive rational numbers

Journal de théorie des nombres de Bordeaux

In this paper we prove a lower bound for the linear dependence of three positive rational numbers under certain weak linear independence conditions on the coefficients of the linear forms. Let $\Lambda ={b}_{2}log{\alpha }_{2}-{b}_{1}log{\alpha }_{1}-{b}_{3}log{\alpha }_{3}\ne 0$ with ${b}_{1},{b}_{2},{b}_{3}$ positive integers and ${\alpha }_{1},{\alpha }_{2},{\alpha }_{3}$ positive multiplicatively independent rational numbers greater than $1$. Let ${\alpha }_{j1}={\alpha }_{j1}/{\alpha }_{j2}$ with ${\alpha }_{j1},{\alpha }_{j2}$ coprime positive integers $\left(j=1,2,3\right)$. Let ${\alpha }_{j}\ge \phantom{\rule{4.0pt}{0ex}}\text{max}\left\{{\alpha }_{j1},e\right\}$ and assume that gcd$\left({b}_{1},{b}_{2},{b}_{3}\right)=1.$ Let ${b}^{\text{'}}=\left(\frac{{b}_{2}}{log{\alpha }_{1}}+\frac{{b}_{1}}{log{\alpha }_{2}}\right)\phantom{\rule{0.166667em}{0ex}}\left(\frac{{b}_{2}}{log{\alpha }_{3}}+\frac{{b}_{3}}{log{\alpha }_{2}}\right)$ and assume that $B\ge \phantom{\rule{4.0pt}{0ex}}\text{max}\left\{10,log{b}^{\text{'}}\right\}.$ We prove that either $\left\{{b}_{1},{b}_{2},{b}_{3}\right\}$ is $\left({c}_{4},B\right)$-linearly dependent over $ℤ$ (with respect to $\left({a}_{1},{a}_{2},{a}_{3}\right)$) or ...

### Enumerating ${A}_{3}\left(2\right)$ blueprints, and an application.

Experimental Mathematics

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