\relax 
\providecommand\hyper@newdestlabel[2]{}
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
\global\let\oldcontentsline\contentsline
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
\global\let\oldnewlabel\newlabel
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
\AtEndDocument{\ifx\hyper@anchor\@undefined
\let\contentsline\oldcontentsline
\let\newlabel\oldnewlabel
\fi}
\fi}
\global\let\hyper@last\relax 
\gdef\HyperFirstAtBeginDocument#1{#1}
\providecommand*\HyPL@Entry[1]{}
\citation{MSS2011}
\citation{AS2000}
\HyPL@Entry{0<</S/D>>}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction and main results}{1}{section.1}}
\newlabel{ex:7.10}{{1}{2}{$M_{10,7}=13$}{theorem.1}{}}
\newlabel{ex:11.10}{{2}{2}{$M_{10,11}=39$}{theorem.2}{}}
\newlabel{definition:Mqn}{{3}{2}{}{theorem.3}{}}
\newlabel{eq:r}{{1}{2}{}{equation.1.1}{}}
\newlabel{them:main}{{4}{2}{}{theorem.4}{}}
\newlabel{eq:leastexample}{{2}{3}{}{equation.1.2}{}}
\newlabel{eq:congruence}{{3}{3}{}{equation.1.3}{}}
\newlabel{them:main2}{{5}{3}{}{theorem.5}{}}
\@writefile{toc}{\contentsline {section}{\numberline {2}Proofs}{3}{section.2}}
\newlabel{lem:tongyu}{{7}{3}{}{theorem.7}{}}
\newlabel{lem:shushu}{{8}{3}{}{theorem.8}{}}
\newlabel{lem:carry}{{9}{4}{}{theorem.9}{}}
\newlabel{lem:cover}{{10}{4}{}{theorem.10}{}}
\newlabel{lem:all}{{11}{5}{}{theorem.11}{}}
\newlabel{eq:two-parts}{{4}{5}{Proofs}{equation.2.4}{}}
\newlabel{eq:modalpha}{{5}{6}{Proofs}{equation.2.5}{}}
\newlabel{eq:modbeta}{{6}{6}{Proofs}{equation.2.6}{}}
\newlabel{eq:alphabeta}{{7}{6}{Proofs}{equation.2.7}{}}
\newlabel{eq:right}{{8}{6}{Proofs}{equation.2.8}{}}
\newlabel{eq:left}{{9}{6}{Proofs}{equation.2.9}{}}
\newlabel{eq:mod}{{10}{6}{Proofs}{equation.2.10}{}}
\newlabel{eq:q-1}{{11}{6}{Proofs}{equation.2.11}{}}
\newlabel{eq:I-sq-k}{{12}{7}{Proofs}{equation.2.12}{}}
\newlabel{eq:I-sq-m}{{13}{7}{Proofs}{equation.2.13}{}}
\newlabel{eq:length}{{2}{7}{Proofs}{Item.9}{}}
\newlabel{eq:(q-1)timesk+r}{{14}{8}{Proofs}{equation.2.14}{}}
\newlabel{eq:example}{{15}{8}{Proofs}{equation.2.15}{}}
\newlabel{eq:turn-to-equality}{{16}{8}{Proofs}{equation.2.16}{}}
\newlabel{eq:mod-simple}{{17}{8}{Proofs}{equation.2.17}{}}
\newlabel{eq:mod-simple2}{{18}{9}{Proofs}{equation.2.18}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3}Further conclusions and open problems}{9}{section.3}}
\newlabel{cor:2}{{12}{9}{Binary system case}{theorem.12}{}}
\newlabel{cor:10}{{13}{9}{Decimalism case}{theorem.13}{}}
\bibcite{MSS2011}{1}
\bibcite{AS2000}{2}
\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces This table is a detailed explanation about Corollary \ref  {cor:10}, in which we symbolically set $(19^0)_{10}= 1$, and the third and fourth columns of this table are the concrete realizations of Eq.\nobreakspace  {}\textup  {\hbox {\mathsurround \z@ \normalfont  (\ignorespaces \ref  {eq:leastexample}\unskip \@@italiccorr )}}.}}{10}{table.1}}
\newlabel{table1}{{1}{10}{This table is a detailed explanation about Corollary \ref {cor:10}, in which we symbolically set $(19^0)_{10}= 1$, and the third and fourth columns of this table are the concrete realizations of Eq.~\eqref {eq:leastexample}}{table.1}{}}