A closed-form bound for linear forms in three $p$-adic logarithms of rational numbers
We deduce from the $p$-adic kit for three logarithms of the companion note hexagon:2610.00706 a fully explicit bound in closed form. If $\gamma_1,\gamma_2,\gamma_3$ are multiplicatively independent rationals, congruent to $1$ modulo $p$ (modulo $4$ if $p=2$), of heights at most $a_j\log p$ with $a_j\ge1$, and if $|b_j|\le B$, then $v_p(\gamma_1^{b_1}\gamma_2^{b_2}\gamma_3^{b_3}-1)\le a_1a_2a_3\,\mathcal{B}^2\max\{27\,711,\ C_2(p)\max_j a_j\}$, where $\mathcal{B}=\max\{\log B/\log p,\,10\}$ and $C_2(p)=11\,107\,\frac{p}{p-1}\bigl(1+\frac{0.31}{\log p}\bigr)^2$, as soon as…
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