\documentclass[11pt,a4paper]{article} \usepackage[utf8]{inputenc} \usepackage{amsmath,amssymb,amsthm} \usepackage[margin=2.4cm]{geometry} \usepackage{booktabs} \usepackage[hidelinks]{hyperref} \newtheorem{theorem}{Theorem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{remark}[theorem]{Remark} \title{The exponent of $p$ in $x^{p-1}-1$:\\ Yamada's constant $283$ lowered to $69$} \author{David Niedbala Giraudin\\\small ORCID 0009-0009-1526-1178} \date{} \renewcommand{\labelitemi}{$\bullet$} \begin{document} \maketitle \begin{abstract} Yamada (J.~Number Theory 130, 2010) deduced from the method of interpolation determinants of Bugeaud and Laurent the bound $v_p(x^{p-1}-1)\le 283\,(p-1)\log y\,\log(xy)/(\log p)^2+4$, for multiplicatively independent integers $x,y$ coprime to~$p$. We replace $283$ by~$69$. The new ingredient is an elementary bound, by rearrangement, on the spread of the exponents that occur in the Liouville inequality: it at least halves the height term of Yamada's criterion, involves no congruence class, and allows the parameters to be re-optimised. The proof is complete up to the lemmas of Bugeaud--Laurent (zero lemma, analytic upper bound) and of Laurent--Mignotte--Nesterenko (bound for the polynomial), which are used as they stand. A self-contained verifier checks the numerical inequalities. \end{abstract} \section{Result} For a prime $p$ and an integer $x$ coprime to $p$, the exponent $v_p(x^{p-1}-1)$ equals $1$ unless $p$ is a Wieferich prime to base $x$. The trivial bound $v_p(x^{p-1}-1)<(p-1)\log x/\log p$ comes from $p^{v}\le x^{p-1}-1$. The theorems of Bugeaud--Laurent \cite{BL} on linear forms in two $p$-adic logarithms assume $\log A_i\ge \log p$ and give nothing better here. Yamada \cite{Y} removed this obstacle and obtained a non-trivial bound, with the constant $283$. \begin{theorem}\label{thm:main} Let $p$ be a prime and $x,y\ge2$ two multiplicatively independent integers, coprime to $p$. Then \[ v_p\bigl(x^{p-1}-1\bigr)\;\le\;69\,(p-1)\,\frac{\log y}{\log p}\,\frac{\log xy}{\log p}\;+\;4 . \] \end{theorem} \begin{corollary}\label{cor} For every odd prime $p$ and every odd prime $q\ne p$, \[ v_p\bigl(q^{p-1}-1\bigr)\le 69\,(p-1)\,\frac{\log 2\,\log 2q}{(\log p)^2}+4, \qquad v_p\bigl(2^{p-1}-1\bigr)\le 69\,(p-1)\,\frac{\log 3\,\log 6}{(\log p)^2}+4 \quad(p\ne 2,3). \] \end{corollary} The corollary follows with $(x,y)=(q,2)$ and $(x,y)=(2,3)$. Theorem~4 of \cite{Y}, on $v_p(\sigma(q^c))$, is deduced from Theorem~2 of \cite{Y} by an inequality of Nagell, valid for $p\notin\{2,q\}$ (and $p\ne2,3$ for $\sigma(2^c)$); in that range it therefore holds with $69$ in place of $283$. \begin{table}[h] \centering\small \begin{tabular}{@{}lcc@{}} \toprule & constant & additive term\\ \midrule Yamada 2010 \cite{Y}, Theorem 2 & $283$ & $4$\\ Theorem \ref{thm:main} & $69$ & $4$\\ \bottomrule \end{tabular} \end{table} Theorem \ref{thm:main} gives nothing when $\log p\le69\,\log y\,\log(xy)/\log x$, since the trivial bound is then better. For $y=2$ and $x$ large it becomes non-trivial as soon as $p>2^{69(1+o(1))}$, against $p>2^{283(1+o(1))}$ in \cite{Y}. \section{A rearrangement lemma} Fix integers $K,L\ge1$, put $N=KL$ and $l_j=\lfloor (j-1)/K\rfloor$ for $1\le j\le N$: each value $l\in\{0,\dots,L-1\}$ is taken exactly $K$ times. \begin{lemma}[rearrangement]\label{lem:R} Let $R\ge1$ and let $r_1,\dots,r_N$ be real numbers in $[0,R-1]$. Put $M=\frac{L-1}{2}\sum_j r_j$. For every permutation $\sigma$ of $\{1,\dots,N\}$, \[ \Bigl|\sum_{i=1}^N l_i\,r_{\sigma(i)}-M\Bigr|\;\le\;G:=\frac{(R-1)\,K\,\lfloor L^2/4\rfloor}{2}. \] The bound is attained when $L$ is even, half of the $r_j$ being $0$ and the other half $R-1$. \end{lemma} \begin{proof} Put $\bar l=(L-1)/2$. Since $\sum_i(l_i-\bar l)=K\sum_{l=0}^{L-1}(l-\bar l)=0$, \[ \sum_i l_i r_{\sigma(i)}-M=\sum_i (l_i-\bar l)\,r_{\sigma(i)} =\sum_i (l_i-\bar l)\Bigl(r_{\sigma(i)}-\frac{R-1}{2}\Bigr), \] whose absolute value is at most $\frac{R-1}{2}\sum_i|l_i-\bar l| =\frac{R-1}{2}\,K\sum_{l=0}^{L-1}\bigl|l-\tfrac{L-1}{2}\bigr|=\frac{R-1}{2}\,K\lfloor L^2/4\rfloor$. In the configuration indicated, the increasing matching gives equality. \end{proof} Lemma~10 of \cite{BL} bounds the same quantity by $\frac{NL(R+g-1)}{8}-\frac{gN^2L}{24(S+g-1)}$, for pairs $(r_j,s_j)$ in a single class $m_1r+m_2s\equiv c \pmod g$. Lemma~6 of \cite{Y} replaces it by $G_Y=NL(R-1)/4$. Lemma~\ref{lem:R} gives $G=NL(R-1)/8$ for $L$ even and $G=NL(R-1)(1-L^{-2})/8$ for $L$ odd, with no hypothesis on the $r_j$. \section{The improved criterion} For $L\ge1$ put \[ \lambda_L=\frac{2\lfloor L^2/4\rfloor}{L}\qquad\Bigl(\lambda_L=\tfrac L2 \text{ if $L$ is even},\quad \lambda_L=\tfrac{L^2-1}{2L}\text{ if $L$ is odd}\Bigr). \] \begin{theorem}\label{thm:crit} Let $p$ be an odd prime, $\alpha_1,\alpha_2$ non-zero integers coprime to $p$, $b_1,b_2$ positive integers with $\gcd(b_1,b_2,p)=1$, and $\Lambda=\alpha_1^{b_1}-\alpha_2^{b_2}$. Let $\zeta$ be a primitive $(p-1)$-th root of unity in $\mathbb Z_p$, $g=p-1$, and $m_i$ integers such that $\alpha_i\equiv\zeta^{m_i}\pmod p$. Let $K\ge3$, $L\ge2$, $R_1,R_2,S_1,S_2\ge1$ be integers, $R=R_1+R_2-1$, $S=S_1+S_2-1$, $N=KL$ and \[ b=\frac{(R-1)b_2+(S-1)b_1}{2}\Bigl(\prod_{j=1}^{K-1}j!\Bigr)^{-2/(K^2-K)} . \] Assume that there are classes $c_1,c_2$ modulo $g$ such that \begin{align} \#\{\alpha_1^r\alpha_2^s : 0\le r (K-1)L,\label{eq:c2} \end{align} and that \begin{equation}\label{eq:crit} K(L-1)\log p\;>\;3\log N+(K-1)\log b+\lambda_L\bigl((R-1)\log|\alpha_1|+(S-1)\log|\alpha_2|\bigr). \end{equation} Then $v_p(\Lambda)\le KL-1$. \end{theorem} Theorem~1 of \cite{Y} has the same form, with the coefficient $L$ in place of $\lambda_L$ in \eqref{eq:crit} (and any integer $g$ with $\alpha_i^g\equiv1$). \begin{proof} This is the proof of Theorem~1 of \cite{BL}, in the case $\mathbb Q_p$: $D=e=f=1$, $t=0$ because $p$ is odd, and $u=0$ because $\gcd(b_1,b_2,p)=1$. Only the arithmetic lower bound changes. Assume $v_p(\Lambda)\ge N-\tfrac12$. The matrix $\mathcal M$ formed by the columns $(r,s)$, $0\le r P_0(x,y):=69\,\frac{\log y\,\log(xy)}{\log x}\ \ \ge\ 69\log 2 . \end{equation} In particular $p>2^{69}$ is odd. We apply Theorem~\ref{thm:crit} with \[ (\alpha_1,\alpha_2)=(xy,y),\qquad b_1=b_2=g=p-1,\qquad \Lambda=(xy)^{p-1}-y^{p-1}=y^{p-1}\bigl(x^{p-1}-1\bigr), \] so that $v_p(\Lambda)=v_p(x^{p-1}-1)$. Write $a_1=\log(xy)/P$, $a_2=\log y/P$, $\Gamma=g\,a_1a_2$, and take \[ L=5,\quad k=\tfrac{69}{25},\quad K=\lfloor kL\Gamma\rfloor+1,\quad R_1=\Bigl\lfloor\sqrt{gLa_2/a_1}\Bigr\rfloor+1,\quad S_1=\Bigl\lfloor\sqrt{gLa_1/a_2}\Bigr\rfloor+1, \] \[ R_2=\Bigl\lfloor\sqrt{g(K-1)La_2/a_1}\Bigr\rfloor+1,\qquad S_2=\Bigl\lfloor\sqrt{g(K-1)La_1/a_2}\Bigr\rfloor+1 . \] Then $R_1S_1>gL$ and $R_2S_2>g(K-1)L$. The classes modulo $g$ partition the boxes, so there is a class $c_1$ (resp. $c_2$) containing at least $L$ (resp. more than $(K-1)L$) points of the box $R_1\times S_1$ (resp. $R_2\times S_2$). The numbers $\alpha_1^r\alpha_2^s$ are pairwise distinct by multiplicative independence, whence \eqref{eq:c1}. Since $xy\ge6$ and $y\ge2$, under \eqref{eq:zone} we have $\Gamma\ge (e^{69\log2}-1)\log2\log6/(69\log 2)^2>3.2\cdot10^{17}$. \subsection{Degenerate case} If $(r,s)\mapsto (p-1)(r+s)$ is not injective on the class $c_2$ of the box $R_2\times S_2$, two points $(r_1,s_1)\ne(r_2,s_2)$ of this class satisfy $r_1+s_1=r_2+s_2$, whence $(m_1-m_2)(r_1-r_2)\equiv0\pmod g$. Since $\zeta^{m_1-m_2}\equiv x$, the order $g_0$ of $x$ modulo $p$ divides $r_1-r_2$, so $g_0kL\Gamma>4\cdot10^{18}$, we have $\eta(K)<10^{-8}$. We conclude with $g0}$ is $e^{E-1}4.5$. We have $h(0)=A$, and $(1+t)^2h'(t)=\frac{2(1+t)}{1+2t}-A-\log(1+2t)\le2-A<0$. Hence $h(t)\le A$, and $u\ge\log2$ gives the bound. \end{proof} \noindent\emph{Verification of \eqref{eq:crit}.} We have \[ (R-1)\log|\alpha_1|=(R-1)a_1P\le P\Bigl(\sqrt{L\Gamma}+\sqrt{g(K-1)La_1a_2}\Bigr)\le P\bigl(\sqrt{L\Gamma}+L\Gamma\sqrt k\bigr), \] and likewise for $(S-1)\log|\alpha_2|$. Also $K-1\le kL\Gamma$, $K\ge kL\Gamma$, $\log b\le\beta^*P$, and $N\le kL^2\Gamma+L$. It therefore suffices that \[ kL(L-1)\ >\ kL\beta^*+2\lambda_5L\sqrt k+2\lambda_5\sqrt{L/\Gamma}+\frac{3\log(kL^2\Gamma+L)}{\Gamma P}. \] With $L=5$, $k=69/25$ and $\lambda_5=12/5$: \[ kL(L-1-\beta^*)=40.0919\ldots,\qquad 2\lambda_5L\sqrt k=39.8717\ldots, \] and the last two terms are less than $2\cdot10^{-8}$ since $\Gamma>3.2\cdot10^{17}$ and $P>47$. The margin is $0.2201$. Theorem~\ref{thm:crit} gives $v_p(\Lambda)\le KL-1\le kL^2\Gamma+L-1=69\,\Gamma+4$, and $\Gamma=(p-1)\log y\log(xy)/P^2$. \qed \section{Remarks} \begin{remark}[choice of $L$] The same chain, with $\lambda_L$ and the corresponding $\beta^*$, gives the minimal constants: \begin{center}\small \begin{tabular}{@{}lccccccc@{}} \toprule $L$ & 3 & 4 & 5 & 6 & 7 & 8 & 9\\ \midrule coefficient $\lambda_L$ & $75.27$ & $70.09$ & $68.29$ & $84.44$ & $95.04$ & $116.30$ & $132.85$\\ coefficient $L$ of \cite{Y} & $337.75$ & $263.13$ & $282.79$ & $327.95$ & $387.46$ & $457.97$ & $538.12$\\ \bottomrule \end{tabular} \end{center} The choice $L=5$ is the best in the table. The choice $L=4$ gives $71$ with additive term $3$. \end{remark} \begin{remark}[the constant $283$] Equation numbers refer to the published version of \cite{Y}. In \cite[(24)]{Y}, the intermediate bound $\log b\le\log p+\log\log p$ does not hold for $y=2$. It would require $\log\bigl(\frac1{\log 2x}+\frac1{\log2}\bigr)+\frac32-\log2-\frac12\log 11.32\le0$, whereas the left-hand side is positive for $3\le x\le 12\,311\,938$, $x$ not a power of $2$ (it equals $0.287$ at $x=3$). With only the reduction $p>2^{283}$ of \cite{Y}, one finds at $(x,y)=(3,2)$ and $\log p=283\log2$ that $\log b/\log p$ can reach $1.02837>1.027$, and (21) of \cite{Y} is no longer satisfied. Statement (6) of \cite{Y} nevertheless remains true. On the one hand, Lemma~\ref{lem:R} gives $G\le NL(R-1)/8\le G_Y$ with no hypothesis; Theorem~1 of \cite{Y} therefore holds even when $\gcd(m_1,m_2,p-1)\ne1$, a case its Lemma~6 does not reach. On the other hand, the chain above, run with the coefficient $L$ of \cite{Y} and the exact range \eqref{eq:zone}, gives the minimal constant $282.79\le283$ at $L=5$. \end{remark} \begin{remark}[additive term] The analytic bound used here (Lemma 8 of \cite{BL}) contains the loss $N\log N/\log p$. Expanding the powers $\theta^{\ell z}$ as a binomial series in $\theta-1$ instead of the exponential series gives $v_p(\Delta)\ge NK(L-1)/2$ with no loss as soon as $v_p(\Lambda)\ge N-1$ \cite[Corollary 3]{NK}; the conclusion of Theorem~\ref{thm:crit} becomes $v_p(\Lambda)\le KL-2$, and the additive term $4$ of Theorem~\ref{thm:main} can be replaced by $3$. \end{remark} \begin{remark}[large $p$] Stewart \cite[proof of Lemma 8, (41)]{St}, using the refinements of Yu \cite{Yu13}, proves, for an integer $a\ge2$ and $p\nmid a$ larger than an effectively computable but unspecified constant, \[ \operatorname{ord}_p(a^{n}-1)