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\title{A closed-form bound for linear forms in three $p$-adic logarithms\\ of rational numbers}
\author{David Niedbala Giraudin\\\small ORCID 0009-0009-1526-1178}
\date{23 October 2026; English version, 30 October 2026}

\begin{document}
\maketitle

\begin{abstract}
We deduce from the $p$-adic kit for three logarithms of \cite{K} 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
\[
v_p\bigl(\gamma_1^{b_1}\gamma_2^{b_2}\gamma_3^{b_3}-1\bigr)\le a_1a_2a_3\,\cB^2\max\Bigl\{27\,711,\ C_2(p)\max_ja_j\Bigr\},
\qquad \cB=\max\Bigl\{\frac{\log B}{\log p},10\Bigr\},
\]
with $C_2(p)=11\,107\,\frac{p}{p-1}\bigl(1+\frac{0.31}{\log p}\bigr)^2$, as soon as
$\gamma_1^{b_1}\gamma_2^{b_2}\gamma_3^{b_3}\ne1$. By a change of basis in the multiplicative group,
the result applies to arbitrary rationals prime to $p$. On the examples of \cite{K}, the bound is
$42$ to $956$ times smaller than that of Yu's explicit theorem, and stays within a factor between
$1.2$ and $2.5$ of the numerical optimisation of the parameters of the kit.
\end{abstract}

\section{Statements}

For a non-zero rational $\gamma=u/w$ written in lowest terms, put $h(\gamma)=\log\max\{|u|,|w|\}$.
We call \emph{principal unit} a rational $\gamma$ prime to $p$ with $\gamma\equiv1\pmod p$ if $p$ is
odd, and $\gamma\equiv1\pmod 4$ if $p=2$.

\begin{theorem}\label{thm:main}
Let $p$ be a prime, $\gamma_1,\gamma_2,\gamma_3$ multiplicatively independent principal units, and
$h_1,h_2,h_3$ real numbers with $h_j\ge\max\{h(\gamma_j),\log p\}$. Put $a_j=h_j/\log p$ $(\ge1)$.
Let $b\in\mathbb Z^3$, $B\ge\max\{1,|b_1|,|b_2|,|b_3|\}$ and $\cB=\max\{\log B/\log p,10\}$. If
$\Lambda=\gamma_1^{b_1}\gamma_2^{b_2}\gamma_3^{b_3}-1\ne0$, then
\[
v_p(\Lambda)\le a_1a_2a_3\,\cB^2\max\bigl\{C_1,\ C_2(p)\max\{a_1,a_2,a_3\}\bigr\},
\]
with $C_1=27\,711$ and $C_2(p)=11\,107\,\frac p{p-1}\bigl(1+\frac{0.31}{\log p}\bigr)^2$.
\end{theorem}

Numerically, $C_2(2)\le46\,528$, $C_2(3)\le27\,390$, $C_2(5)\le19\,748$, $C_2(7)\le17\,416$,
$C_2(101)\le12\,776$, and $C_2(p)\to11\,107$. The term $C_1$ comes from the main case, the term
$C_2(p)\max a_j$ from the degenerate cases.

For arbitrary rationals, one reduces to the theorem by a change of basis. Let
$\alpha_1,\alpha_2,\alpha_3$ be non-zero rationals prime to $p$, multiplicatively independent. For
$p$ odd, let $\zeta$ be a primitive $(p-1)$-th root of unity in $\mathbb Z_p$ and $n_i$ the
Teichm\"uller index of $\alpha_i$ ($\alpha_i\equiv\zeta^{n_i}\pmod p$); put $g=p-1$. For $p=2$, put
$g=2$, $n_i=0$ if $\alpha_i\equiv1\pmod4$ and $n_i=1$ otherwise. Let
\[
\Lambda_0=\{x\in\mathbb Z^3:n_1x_1+n_2x_2+n_3x_3\equiv0\pmod g\},
\]
$e_1,e_2,e_3$ a basis of $\Lambda_0$, $E$ the matrix with rows $e_j$,
$\gamma_j=\alpha_1^{e_{j1}}\alpha_2^{e_{j2}}\alpha_3^{e_{j3}}$ and $c_j=\sum_k|(E^{-1})_{kj}|$.

\begin{corollary}\label{cor:main}
With these notations, let $h_j\ge\max\{h(\gamma_j),\log p\}$, $a_j=h_j/\log p$, $b\in\mathbb Z^3$ with
$|b_i|\le B$ and $\alpha_1^{b_1}\alpha_2^{b_2}\alpha_3^{b_3}\ne1$. Put
$\cB'=\max\{\log(B\max_jc_j)/\log p,10\}$. Then
\[
v_p\bigl(\alpha_1^{b_1}\alpha_2^{b_2}\alpha_3^{b_3}-1\bigr)\le a_1a_2a_3\,\cB'^2\max\Bigl\{C_1,\ C_2(p)\max_ja_j\Bigr\}.
\]
\end{corollary}

Since $\Lambda=\alpha_3^{b_3}\bigl(\alpha_1^{b_1}\alpha_2^{b_2}\alpha_3^{-b_3}-1\bigr)$, the
corollary also applies to $\Lambda=\alpha_1^{b_1}\alpha_2^{b_2}-\alpha_3^{b_3}$.

\begin{remark}
The bound is in $(\log B)^2$, whereas that of Yu \cite{Yu} is in $\log B$; like the kit, it is
therefore meant for the reduction of bounds. In the example $p=7$, $\alpha=(2,3,5)$ of
Section~\ref{sec:ex}, it remains better than the explicit form of Yu given in \cite{AKMRVW} as long as
$\log B<1.77\cdot10^4$, that is $B<10^{7711}$. It contains no factor $p$: for principal units, none
appears in Corollary~2 of \cite{BL} either. For arbitrary rationals, it moves into the heights. For
instance, if the numerators and denominators of the $\alpha_i$ are pairwise coprime, then
$h(\alpha^e)\ge\frac12\sum_i|e_i|h(\alpha_i)$ for $e\in\mathbb Z^3$, and Hadamard's inequality gives
$h_1h_2h_3\ge\frac{[\mathbb Z^3:\Lambda_0]}8h(\alpha_1)h(\alpha_2)h(\alpha_3)$; without this
coprimality hypothesis, this lower bound may fail.
\end{remark}

\begin{remark}
The constants are dictated by the case $\cB=10$. For large $\cB$, if $\ell$ is allowed to depend on
$\cB$ (instead of being fixed at $7.2$), the same construction gives, in the main case, a constant
tending to $520\,\ell_\infty^2\approx1.39\cdot10^4$, where $\ell_\infty=2/(1-\theta_\infty)$ and
$\theta_\infty$ is the limit of $\theta(L)$ defined in Section~\ref{sec:param}.
\end{remark}

\section{Results used}

\subsection*{The kit with $g=1$}
The following result is Theorem~11 of \cite{K}, that is, its Theorem~2 with $g=1$; the exceptional
cases (2.14) and (2.15) are those of Mignotte and Voutier \cite{MV}. Let $\gamma_1,\gamma_2,\gamma_3$
be multiplicatively independent principal units, $h_j\ge h(\gamma_j)$, and
$\Lambda'=\gamma_1^{\beta_1}\gamma_2^{\beta_2}-\gamma_3^{\beta_3}\ne0$ with $\beta_j\ge1$,
$\gcd(\beta_1,\beta_2,\beta_3)=1$, $p\nmid\beta_3$ and $\beta_j\le B$. Write
$d_1=\gcd(\beta_1,\beta_3)$, $d_2=\gcd(\beta_2,\beta_3)$,
\[
F_K=\Bigl(\prod_{k=1}^{K-1}(k!)^{K-k}\Bigr)^{-12/(K^3-K)},\qquad
b=\frac{(R-1)\beta_3+(T-1)\beta_1}{2d_1}\cdot\frac{(S-1)\beta_3+(T-1)\beta_2}{2d_2}\cdot F_K,
\]
$E(K,L)=(2K+1)(L-1)/3$, $\lambda_L=2\lfloor L^2/4\rfloor/L$ and $N=K(K+1)L/2$.

\begin{theorem}[\cite{K}]\label{thm:kit}
Let $K\ge2$, $L\ge2$, $R_i,S_i,T_i\ge1$ $(i=1,2,3)$ be integers, $R=R_1+R_2+R_3+1$,
$S=S_1+S_2+S_3+1$, $T=T_1+T_2+T_3+1$, $\chi>0$, $V_1=(R_1+1)(S_1+1)(T_1+1)$ and
$\cM=\max\{R_1+S_1+1,S_1+T_1+1,R_1+T_1+1,\chi\sqrt{V_1}\}$. Assume
\begin{align*}
(\mathrm{Z}1)\quad &V_1>K\cM\quad\text{and}\quad V_1>L,\\
(\mathrm{Z}2)\quad &(R_2+1)(S_2+1)(T_2+1)>\max\{2KL,K^2\},\\
(\mathrm{Z}3)\quad &(R_3+1)(S_3+1)(T_3+1)>3K^2L,
\end{align*}
\begin{equation}\label{eq:crit}
E(K,L)\log p>\log N+\frac{2(K-1)}3\log b+\lambda_L\bigl((R-1)h_1+(S-1)h_2+(T-1)h_3\bigr).
\end{equation}
Then $v_p(\Lambda')\le KL-2$, except in one of the following two cases:
\begin{itemize}
\item[(2.14)] $\beta_1\le\max\{R_1,R_2\}$, $\beta_2\le\max\{S_1,S_2\}$, $\beta_3\le\max\{T_1,T_2\}$;
then $v_p(\Lambda')\le(\log2+\beta_1h_1+\beta_2h_2+\beta_3h_3)/\log p$;
\item[(2.15)] there are coprime integers $(u_1,u_2,u_3)\ne0$ with $u_1\beta_1+u_2\beta_2+u_3\beta_3=0$
and $|u_i|\le U_i$, where
\[
U_1=\frac{(S_1+1)(T_1+1)}{\cM-\max\{S_1,T_1\}},\quad
U_2=\frac{(R_1+1)(T_1+1)}{\cM-\max\{R_1,T_1\}},\quad
U_3=\frac{(R_1+1)(S_1+1)}{\cM-\max\{R_1,S_1\}}.
\]
\end{itemize}
\end{theorem}

\subsection*{Two logarithms}

\begin{theorem}[{\cite[Corollaire~2]{BL}}]\label{thm:BL}
Let $\gamma,\gamma'$ be multiplicatively independent principal units, $c,c'\ge1$ integers,
$H\ge\max\{h(\gamma),\log p\}$, $H'\ge\max\{h(\gamma'),\log p\}$ and $b^\star=c/H'+c'/H$. Then
\[
v_p\bigl(\gamma^c-\gamma'^{c'}\bigr)\le\frac{24p}{(p-1)(\log p)^4}
\Bigl(\max\bigl\{\log b^\star+\log\log p+0.4,\ 10\log p,\ 10\bigr\}\Bigr)^2HH'.
\]
\end{theorem}

For $p=2$, the integer $t$ of \cite{BL} equals $1$ and the hypothesis $\gamma,\gamma'\in U_v^1$ is
satisfied.

\subsection*{A lemma on factorials}

\begin{lemma}\label{lem:F}
For every integer $K\ge2$, $\log F_K\le-2\log K+\frac{11}3+\frac{4\log K}K$.
\end{lemma}

\begin{proof}
We have $\sum_{k=1}^{K-1}(K-k)\log k!=\sum_{j=1}^Kw_j\log j$ with $w_j=\frac{(K-j)(K-j+1)}2$. Let
$w(x)=\frac{(K-x)(K-x-1)}2$. For $2\le j\le K-1$ and $x\in[j-1,j]$, we have $0\le w(x)\le w_j$ and
$0\le\log x\le\log j$; on $[0,1]$, $w\ge0$ and $\log x\le0$; on $[K-1,K]$, $w\le0$ and $\log x>0$,
while $w_1\log1=w_K\log K=0$. Hence
\[
\sum_{j=1}^Kw_j\log j\ \ge\ \int_0^Kw(x)\log x\,dx=\frac{K^3}6\Bigl(\log K-\frac{11}6\Bigr)-\frac{K^2}4\Bigl(\log K-\frac32\Bigr),
\]
using $\int_0^1(1-t)^n\log t\,dt=-H_{n+1}/(n+1)$, where $H_{n+1}$ is the harmonic number. Thus
$-\log F_K\ge G_K:=\bigl(2K^3\log K-\frac{11}3K^3-3K^2\log K+\frac92K^2\bigr)/(K^3-K)$, and
\[
G_K-\Bigl(2\log K-\frac{11}3-\frac{4\log K}K\Bigr)
=\frac{K^2\log K+\frac92K^2+2K\log K-\frac{11}3K-4\log K}{K^3-K}>0
\]
for $K\ge2$.
\end{proof}

\section{Choice of the parameters}\label{sec:param}

We fix $m=520$, $\kappa=0.65$, $\ell=7.2$. Let $a_1,a_2,a_3\ge1$, $A=a_1a_2a_3$, $P=\log p$,
$\cB\ge10$, and put
\[
L=\lceil\ell\cB\rceil,\quad K=\lfloor mLA\rfloor,\quad \chi=\kappa\sqrt L,\quad
c_1=(m\kappa)^{2/3}L,\quad c_2=(mL)^{2/3},\quad c_3=(3m^2)^{1/3}L,
\]
\[
R_i=\lfloor c_ia_2a_3\rfloor,\quad S_i=\lfloor c_ia_1a_3\rfloor,\quad T_i=\lfloor c_ia_1a_2\rfloor\qquad(i=1,2,3).
\]
We have $L\ge L_0=72$ and $K\ge K_0=\lfloor mL_0\rfloor=37\,440$. Put
$\sigma=c_1+c_2+c_3=\tau(L)\,mL$, where
\[
\tau(L)=\kappa^{2/3}m^{-1/3}+(mL)^{-1/3}+(3/m)^{1/3},\qquad
\theta(L,K)=\frac94\,\tau(L)\Bigl(1+\frac1K\Bigr).
\]
These quantities decrease in $L$ and $K$; we have $\tau(L_0)\le0.30256$ and
$\theta(L,K)\le\theta_0=\theta(L_0,K_0)\le0.68078$. Finally $R_i+1>c_ia_2a_3$, so that
\begin{equation}\label{eq:box}
(R_i+1)(S_i+1)(T_i+1)>c_i^3A^2,\qquad R_i\le c_ia_2a_3,\ S_i\le c_ia_1a_3,\ T_i\le c_ia_1a_2 .
\end{equation}

\begin{lemma}\label{lem:Z}
We have $K\ge2$, $L\ge2$, $R_i,S_i,T_i\ge1$, and conditions (Z1), (Z2), (Z3) hold.
\end{lemma}

\begin{proof}
The $c_i$ are $\ge1$ and $a_j\ge1$. Since $K\le mLA$ and $A\ge1$, \eqref{eq:box} gives
$\sqrt{V_1}>c_1^{3/2}A=\kappa\sqrt L\cdot mLA\ge\chi K$, hence $V_1>K\chi\sqrt{V_1}$. Next
$R_1+S_1+1\le(2c_1+1)A\le3c_1A$, and $V_1>c_1^3A^2\ge3mLc_1A^2\ge K(R_1+S_1+1)$ because
$c_1^2=(m\kappa)^{4/3}L^2\ge3mL$, which is equivalent to $m^{1/3}\kappa^{4/3}L\ge3$ (the left-hand
side is at least $326$); likewise for $S_1+T_1+1$ and $R_1+T_1+1$. Hence $V_1>K\cM$, and
$V_1>c_1^3\ge L$. For (Z2), $c_2^3A^2=m^2L^2A^2\ge K^2$ and $m^2L^2A^2\ge2mL^2A^2\ge2KL$. For
(Z3), $c_3^3A^2=3m^2L^3A^2\ge3K^2L$.
\end{proof}

\begin{lemma}\label{lem:crit}
Let $h_j=a_jP$ and $\beta_j\le B$ with $\log B\le\cB P$. Criterion \eqref{eq:crit} holds.
\end{lemma}

\begin{proof}
\emph{Height term.} By \eqref{eq:box}, $R-1=R_1+R_2+R_3\le\sigma a_2a_3$, whence
$(R-1)h_1\le\sigma AP$, and likewise for $S$, $T$. Since $\lambda_L\le L/2$ and
$A\le(K+1)/(mL)$,
\[
\lambda_L\bigl((R-1)h_1+(S-1)h_2+(T-1)h_3\bigr)\le\frac32L\sigma AP\le\frac32\tau(L)L(K+1)P.
\]
\emph{Term $\log b$.} We have $(R-1)\beta_3+(T-1)\beta_1\le(R+T-2)B\le\sigma(a_2a_3+a_1a_2)B\le2\sigma AB$,
and likewise for the second factor; with Lemma~\ref{lem:F},
$\log b\le2\log B+2\log(\sigma A/K)+\frac{11}3+\frac{4\log K}K$. Since
$K\ge mLA(1-\frac1{mL})$, we have $\sigma A/K\le\tau(L)/(1-\frac1{mL})$, hence $\log b\le2\log B+\beta$
with
\[
\beta=2\log\tau(L)-2\log\Bigl(1-\frac1{mL}\Bigr)+\frac{11}3+\frac{4\log K}K\le\beta_0\le1.27688,
\]
$\beta_0$ being the value at $(L_0,K_0)$ (each term decreases). Thus
$\frac{2(K-1)}3\log b\le\frac{2K}3(2\log B+\beta_0)$, since $\beta_0>0$.

\emph{Term $\log N$.} $\log N\le\log K+\log(K+1)+\log L\le3\log(K+1)$.

\emph{Conclusion.} Since $E(K,L)\ge\frac{2K}3(L-1)$, it suffices, after division by $\frac{2K}3P$, that
\[
L\bigl(1-\theta(L,K)\bigr)>1+\frac{2\log B}P+\frac1P\Bigl(\beta_0+\frac{9\log(K+1)}{2K}\Bigr).
\]
We have $L(1-\theta(L,K))\ge\ell\cB(1-\theta_0)$, $2\log B/P\le2\cB$, $P\ge\log2$ and
$\frac{9\log(K+1)}{2K}\le\frac{9\log(K_0+1)}{2K_0}$. It therefore suffices that
$\cB\bigl(\ell(1-\theta_0)-2\bigr)>1+\bigl(\beta_0+\frac{9\log(K_0+1)}{2K_0}\bigr)/\log2$, that is
$0.2983\,\cB>2.8440$, which holds for $\cB\ge10$.
\end{proof}

\begin{lemma}\label{lem:U}
We have $\cM>\max\{R_1,S_1,T_1\}$ and $U_j\le\mu\eta\,a_j$ for $j=1,2,3$, where
\[
\mu=m^{1/3}\kappa^{-2/3}\le10.7167,\qquad
\eta=\frac{(1+1/c_1(L_0))^2}{1-1/(\kappa^{4/3}m^{1/3}L_0)}\le1.00366.
\]
\end{lemma}

\begin{proof}
Put $x_1=c_1a_2a_3$, $x_2=c_1a_1a_3$, $x_3=c_1a_1a_2$, so that $R_1\le x_1<R_1+1$, etc. We have
$\cM\ge\chi\sqrt{V_1}>\chi c_1^{3/2}A$ and $\max\{S_1,T_1\}\le c_1a_1\max\{a_2,a_3\}\le c_1A$, hence
$\cM-\max\{S_1,T_1\}>c_1A(\chi\sqrt{c_1}-1)>0$. Next
$(S_1+1)(T_1+1)\le x_2x_3(1+1/c_1)^2=c_1^2a_1A(1+1/c_1)^2$. Hence
\[
U_1\le\frac{c_1a_1(1+1/c_1)^2}{\chi\sqrt{c_1}-1}=\frac{\sqrt{c_1}}\chi\,a_1\,\frac{(1+1/c_1)^2}{1-1/(\chi\sqrt{c_1})},
\]
with $\sqrt{c_1}/\chi=\mu$, $\chi\sqrt{c_1}=\kappa^{4/3}m^{1/3}L\ge\kappa^{4/3}m^{1/3}L_0$ and
$c_1\ge c_1(L_0)$. Likewise for $U_2$, $U_3$.
\end{proof}

\section{Proof of Theorem \ref{thm:main}}

\emph{Reduction.} Let $\delta=\gcd(b_1,b_2,b_3)$ and
$y=\gamma_1^{b_1/\delta}\gamma_2^{b_2/\delta}\gamma_3^{b_3/\delta}$. It is a principal unit, $y\ne1$, and
$v_p(\Lambda)=v_p(y^\delta-1)=v_p(y-1)+v_p(\delta)$ (for $p=2$, one uses $y\equiv1\pmod4$). We have
$v_p(\delta)\le\log B/P\le\cB\le A\cB^2/10$. Since $b/\delta$ is primitive, one of its coordinates,
say $j_0$, is prime to $p$. Replacing some $\gamma_j$ by $\gamma_j^{-1}$ (which does not change
$h(\gamma_j)$) and multiplying by a unit, $v_p(y-1)$ is the valuation of
$\gamma_i^{\beta_i}\gamma_{i'}^{\beta_{i'}}-\gamma_{j_0}^{\beta_{j_0}}$, with $\{i,i',j_0\}=\{1,2,3\}$,
$\beta_j=|b_j|/\delta\le B$ and $p\nmid\beta_{j_0}$. We renumber so that $j_0=3$; the statements being
symmetric in $a_1,a_2,a_3$, this does not change the bound.

\emph{One or two zero exponents.} If two of the $\beta_j$ vanish,
$v_p(y-1)=v_p(\gamma_j^\beta-1)=v_p(\gamma_j-1)+v_p(\beta)\le(h_j+\log2+\log B)/P\le a_j+1+\cB$. If
only one vanishes, Theorem~\ref{thm:BL}, applied to the two units $\gamma_j,\gamma_k$ whose exponents
do not vanish, with $H=h_j$ and $H'=h_k$, gives the bound of case (2.15) below with $24$ instead of
$96\mu^2\eta^2$ and $a_ja_k\le A$ instead of $A\max a_j$. In both cases we stay below the stated
bound.

\emph{Three non-zero exponents.} We apply Theorem~\ref{thm:kit} with the parameters of
Section~\ref{sec:param} and $h_j=a_jP$; its hypotheses hold (Lemmas~\ref{lem:Z} and
\ref{lem:crit}). Outside cases (2.14), (2.15), $v_p(y-1)\le KL-2\le mL^2A$, and
$L\le\ell\cB+1\le(\ell+\frac1{10})\cB$, whence
\[
v_p(\Lambda)\le\Bigl(m\bigl(\ell+\tfrac1{10}\bigr)^2+\tfrac1{10}\Bigr)A\cB^2\le27\,711\,A\cB^2.
\]

\emph{Case (2.14).} Since $c_2\le c_1$ (as $L\ge3$), $\beta_1h_1+\beta_2h_2+\beta_3h_3\le3c_1AP$, and
$v_p(y-1)\le1+3c_1A\le1+3\cdot48.53\cdot7.3\,A\cB\le107\,A\cB^2$.

\emph{Case (2.15).} Put $x=\gamma_3^{\beta_3}\gamma_1^{-\beta_1}\gamma_2^{-\beta_2}$, so that
$v_p(y-1)=v_p(x-1)$. We have $x\ne\pm1$, since $\Lambda'\ne0$ and $x$ is a product of powers of the
$\gamma_j$, which are multiplicatively independent. For every integer $u\ne0$, therefore, $x^u\ne1$
and $v_p(x-1)\le v_p(x^u-1)$. If $u_3\ne0$, then $|u_3|\ge1$, hence $U_3\ge1$, and
\[
x^{u_3}=\gamma_1'^{-\beta_1}\gamma_2'^{-\beta_2},\qquad\gamma_1'=\gamma_1^{u_3}\gamma_3^{u_1},\quad
\gamma_2'=\gamma_2^{u_3}\gamma_3^{u_2},
\]
where $\gamma_1',\gamma_2'$ are multiplicatively independent principal units with
$h(\gamma_1')\le H_1=U_3h_1+U_1h_3$ and $h(\gamma_2')\le H_2=U_3h_2+U_2h_3$; we have $H_i\ge P$. If
$u_3=0$, then $u_1u_2\ne0$ and $x^{u_1}=(\gamma_1^{u_2}\gamma_2^{-u_1})^{\beta_2}(\gamma_3^{u_1})^{\beta_3}$,
with heights $H_1=U_2h_1+U_1h_2$ and $H_2=U_1h_3$, both $\ge P$. In both cases, up to a unit and after
possibly inverting $\gamma_1'$ or $\gamma_2'$, we are reduced to $\gamma^c-\gamma'^{c'}$ with
$c,c'\le B$. By Lemma~\ref{lem:U}, $H_1H_2\le4\mu^2\eta^2A\max_ja_jP^2$. In Theorem~\ref{thm:BL},
$b^\star\le2B/P$, hence $\log b^\star+\log\log p+0.4\le\log B+1.1\le\cB P+1.1$; moreover
$10\log p\le\cB P$ and $10\le\cB P+3.1$. Since $\cB\ge10$, the maximum is at most
$\cB P+0.31\cB=\cB P(1+0.31/P)$. Hence
\[
v_p(y-1)\le\frac{24p}{(p-1)P^4}\cB^2P^2\Bigl(1+\frac{0.31}P\Bigr)^2\cdot4\mu^2\eta^2A\max_ja_jP^2
=96\mu^2\eta^2\frac p{p-1}\Bigl(1+\frac{0.31}{\log p}\Bigr)^2A\max_ja_j\,\cB^2.
\]
With $96\mu^2\eta^2\le11\,106$ and $v_p(\delta)\le A\cB^2/10$, we obtain
$v_p(\Lambda)\le C_2(p)A\max_ja_j\,\cB^2$.

In all cases, $v_p(\Lambda)\le A\cB^2\max\{C_1,C_2(p)\max_ja_j\}$. \qed

\section{Proof of Corollary \ref{cor:main}}

If $v_p(\alpha^b-1)=0$, there is nothing to prove; for $p=2$, neither if $v_2(\alpha^b-1)=1$.
Otherwise $\alpha^b\equiv1\pmod p$ (modulo $4$ if $p=2$), so the Teichm\"uller component of
$\alpha^b$, equal to $\zeta^{n\cdot b}$ (to $(-1)^{n\cdot b}$ if $p=2$), is $1$, and $b\in\Lambda_0$.
The $\gamma_j$ have trivial Teichm\"uller component; they are principal units, multiplicatively
independent since $E$ is invertible. Write $b=\sum_jb'_je_j$ with $b'=bE^{-1}\in\mathbb Z^3$, whence
$|b'_j|\le c_jB$ and $\alpha^b=\gamma^{b'}$. Apply Theorem~\ref{thm:main} to $\gamma^{b'}-1$ with
$B'=B\max_jc_j$; we have $B'\ge1$, since $b'\ne0$ (because $\alpha^b\ne1$), so
$1\le\max_j|b'_j|\le B'$. \qed

\section{Examples}\label{sec:ex}

The table takes up the examples of \cite[Section~7]{K}, with $\alpha=(2,3,5)$ if $p=7$ and
$\alpha=(2,3,7)$ otherwise, and bases of $\Lambda_0$ with the same heights and the same $c_j$. The
column ``kit'' is the numerical optimisation of the parameters of \cite{K}; the column ``Yu'' is the
explicit form of \cite{AKMRVW}, applied to the same $\alpha_i$.

For instance, for $p=7$ and $\alpha=(2,3,5)$, the basis $(-3,0,0)$, $(-1,2,0)$, $(-1,1,-1)$ of
$\Lambda_0$ gives $\gamma=(1/8,9/2,3/10)$, $c=(2/3,1,1)$, $a=(1.069;\,1.129;\,1.183)$, and for
$B=10^{30}$, $\cB'=35.5$, whence $v_7\le4.99\cdot10^7$.

The bounds in the table are rounded up; the ratio is their integer part.

\begin{table}[h]
\centering\small
\begin{tabular}{@{}rcccccr@{}}
\toprule
$p$ & $B$ & heights $h_j$ & Corollary \ref{cor:main} & kit \cite{K} & Yu & Yu / Cor.~\ref{cor:main}\\
\midrule
5 & $10^{10}$ & 1.79\ 1.95\ 2.08 & $1.01\cdot10^7$ & $4.62\cdot10^6$ & $3.08\cdot10^9$ & 306\\
5 & $10^{30}$ & 1.79\ 1.95\ 2.08 & $8.94\cdot10^7$ & $3.98\cdot10^7$ & $9.24\cdot10^9$ & 103\\
7 & $10^{10}$ & 2.08\ 2.20\ 2.30 & $5.54\cdot10^6$ & $2.68\cdot10^6$ & $4.27\cdot10^9$ & 771\\
7 & $10^{30}$ & 2.08\ 2.20\ 2.30 & $4.99\cdot10^7$ & $2.11\cdot10^7$ & $1.28\cdot10^{10}$ & 257\\
11 & $10^{10}$ & 2.48\ 2.64\ 3.30 & $4.35\cdot10^6$ & $2.10\cdot10^6$ & $4.16\cdot10^9$ & 956\\
11 & $10^{30}$ & 2.48\ 2.64\ 3.30 & $3.61\cdot10^7$ & $1.60\cdot10^7$ & $1.25\cdot10^{10}$ & 345\\
13 & $10^{10}$ & 2.64\ 2.77\ 3.30 & $3.97\cdot10^6$ & $1.75\cdot10^6$ & $6.37\cdot10^8$ & 160\\
13 & $10^{30}$ & 2.64\ 2.77\ 3.30 & $2.90\cdot10^7$ & $1.24\cdot10^7$ & $1.91\cdot10^9$ & 66\\
101 & $10^{10}$ & 4.68\ 4.85\ 8.88 & $5.69\cdot10^6$ & $2.33\cdot10^6$ & $2.43\cdot10^8$ & 42\\
101 & $10^{30}$ & 4.68\ 4.85\ 8.88 & $1.25\cdot10^7$ & $9.98\cdot10^6$ & $7.29\cdot10^8$ & 58\\
\bottomrule
\end{tabular}
\caption{Upper bounds for $v_p(\alpha_1^{b_1}\alpha_2^{b_2}\alpha_3^{b_3}-1)$ for $|b_i|\le B$.}
\end{table}

\section*{Verification}
The ancillary file \texttt{VERIF\_FERME3P.py} (plain Python, no argument, a few seconds; messages in
French) checks Lemma~\ref{lem:F} for $2\le K\le3000$ and two large values; checks the numerical
values quoted ($\tau(L_0)$, $\theta_0$, $\beta_0$, $\mu$, $\eta$, $96\mu^2\eta^2$, $C_1$, $C_2(p)$);
and checks directly, on a grid of $27\,440$ points ($p$ up to $10^9+7$, $a_j$ up to $77$ in all
orders, $\log B/\log p$ up to $10^5$), conditions (Z1)--(Z3), criterion \eqref{eq:crit}, the
inequality $\cM>\max\{R_1,S_1,T_1\}$, the bounds for the $U_j$ and those of all branches. It
recomputes the ten rows of the table (bases, indices, congruences $\gamma_j\equiv1\pmod p$, $c_j$,
bounds, ratios).

\paragraph{Acknowledgements.} Technical assistance: Claude (Anthropic).

\raggedright
\begin{thebibliography}{9}
\bibitem{AKMRVW} A.~Alvarado, A.~Koutsianas, B.~Malmskog, C.~Rasmussen, C.~Vincent, M.~West,
A robust implementation for solving the $S$-unit equation and several applications,
arXiv:1903.00977.
\bibitem{BL} Y.~Bugeaud, M.~Laurent, Minoration effective de la distance $p$-adique entre
puissances de nombres alg\'ebriques, \emph{J.~Number Theory} \textbf{61} (1996), 311--342.
\bibitem{MV} M.~Mignotte, P.~Voutier, A kit for linear forms in three logarithms, \emph{Math.
Comp.} \textbf{93} (2024), 1903--1951; arXiv:2205.08899.
\bibitem{K} D.~Niedbala Giraudin, A kit for linear forms in three $p$-adic logarithms, preprint,
version 4, 30 October 2026.
\bibitem{Yu} K.~Yu, $p$-adic logarithmic forms and group varieties III, \emph{Forum Math.}
\textbf{19} (2007), 187--280.
\end{thebibliography}

\end{document}