\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{Hardy}
\citation{online}
\citation{online}
\HyPL@Entry{0<</S/D>>}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}{section.1}}
\newlabel{sec:1}{{1}{1}{Introduction}{section.1}{}}
\citation{C}
\citation{CK}
\citation{KD}
\citation{G}
\citation{FILS}
\citation{B1}
\citation{B2}
\@writefile{toc}{\contentsline {section}{\numberline {2}Statements of the main results}{2}{section.2}}
\newlabel{sec:1-bis}{{2}{2}{Statements of the main results}{section.2}{}}
\newlabel{def:1}{{2}{2}{}{theorem.2}{}}
\newlabel{eq:1}{{1}{2}{}{equation.2.1}{}}
\newlabel{def:2}{{3}{2}{}{theorem.3}{}}
\newlabel{eq:2}{{2}{2}{}{equation.2.2}{}}
\newlabel{q:1}{{6}{3}{}{theorem.6}{}}
\newlabel{q:2}{{7}{3}{}{theorem.7}{}}
\newlabel{q:3}{{8}{3}{}{theorem.8}{}}
\newlabel{q:4}{{9}{3}{}{theorem.9}{}}
\newlabel{thm:1}{{10}{3}{}{theorem.10}{}}
\newlabel{eq:3}{{3}{3}{}{equation.2.3}{}}
\newlabel{r:1}{{11}{3}{}{theorem.11}{}}
\newlabel{thm:11}{{12}{3}{}{theorem.12}{}}
\newlabel{eq:3**}{{4}{3}{}{equation.2.4}{}}
\newlabel{thm:larger-growth}{{15}{4}{}{theorem.15}{}}
\newlabel{eq:3larger-growth}{{5}{4}{}{equation.2.5}{}}
\newlabel{q:13}{{23}{5}{}{theorem.23}{}}
\newlabel{r:2}{{24}{5}{}{theorem.24}{}}
\newlabel{thm:niven-not-mrh}{{25}{5}{}{theorem.25}{}}
\newlabel{thm:noARH}{{26}{5}{}{theorem.26}{}}
\newlabel{thm:mrhexample}{{27}{5}{}{theorem.27}{}}
\newlabel{eq:mrhex}{{6}{5}{}{equation.2.6}{}}
\citation{N}
\newlabel{coro:number}{{31}{6}{}{theorem.31}{}}
\newlabel{q:6}{{33}{6}{}{theorem.33}{}}
\newlabel{q:7}{{34}{6}{}{theorem.34}{}}
\newlabel{thm:2}{{35}{6}{}{theorem.35}{}}
\newlabel{thm:2-strong}{{37}{7}{}{theorem.37}{}}
\newlabel{eq:4-Strong}{{7}{7}{}{equation.2.7}{}}
\newlabel{thm:3}{{38}{7}{}{theorem.38}{}}
\newlabel{thm:3-new}{{40}{7}{}{theorem.40}{}}
\newlabel{eq:5-new}{{8}{7}{}{equation.2.8}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3}Proof of Example \ref  {thm:1}}{8}{section.3}}
\newlabel{sec:2}{{3}{8}{Proof of Example \ref {thm:1}}{section.3}{}}
\newlabel{eq:5}{{9}{8}{Proof of Example \ref {thm:1}}{equation.3.9}{}}
\newlabel{eq:6}{{10}{8}{Proof of Example \ref {thm:1}}{equation.3.10}{}}
\@writefile{toc}{\contentsline {section}{\numberline {4}Proof of Theorem \ref  {thm:11}}{8}{section.4}}
\newlabel{sec:2bis}{{4}{8}{Proof of Theorem \ref {thm:11}}{section.4}{}}
\citation{D}
\@writefile{toc}{\contentsline {section}{\numberline {5}Proof of Theorem \ref  {thm:larger-growth}}{9}{section.5}}
\newlabel{sec:3larger-growth}{{5}{9}{Proof of Theorem \ref {thm:larger-growth}}{section.5}{}}
\@writefile{toc}{\contentsline {section}{\numberline {6}Proof of Theorem \ref  {thm:niven-not-mrh}}{9}{section.6}}
\newlabel{sec:niven-not-mrh}{{6}{9}{Proof of Theorem \ref {thm:niven-not-mrh}}{section.6}{}}
\newlabel{eq:contrad}{{12}{10}{Proof of Theorem \ref {thm:niven-not-mrh}}{equation.6.12}{}}
\@writefile{toc}{\contentsline {section}{\numberline {7}Proof of Theorem \ref  {thm:noARH}}{10}{section.7}}
\newlabel{sec:noARH}{{7}{10}{Proof of Theorem \ref {thm:noARH}}{section.7}{}}
\newlabel{eq:new-add1}{{13}{10}{Proof of Theorem \ref {thm:noARH}}{equation.7.13}{}}
\@writefile{toc}{\contentsline {section}{\numberline {8}Proof of Theorem \ref  {thm:mrhexample}}{10}{section.8}}
\newlabel{sec:mrhexample}{{8}{10}{Proof of Theorem \ref {thm:mrhexample}}{section.8}{}}
\@writefile{toc}{\contentsline {section}{\numberline {9}Proof of Theorem \ref  {thm:2}}{11}{section.9}}
\newlabel{sec:3}{{9}{11}{Proof of Theorem \ref {thm:2}}{section.9}{}}
\newlabel{eq:7}{{14}{11}{Proof of Theorem \ref {thm:2}}{equation.9.14}{}}
\newlabel{eq:8}{{15}{11}{Proof of Theorem \ref {thm:2}}{equation.9.15}{}}
\newlabel{eq:9}{{16}{11}{Proof of Theorem \ref {thm:2}}{equation.9.16}{}}
\newlabel{eq:10}{{17}{11}{Proof of Theorem \ref {thm:2}}{equation.9.17}{}}
\newlabel{eq:11}{{18}{11}{Proof of Theorem \ref {thm:2}}{equation.9.18}{}}
\newlabel{eq:12}{{19}{11}{Proof of Theorem \ref {thm:2}}{equation.9.19}{}}
\newlabel{eq:13}{{20}{11}{Proof of Theorem \ref {thm:2}}{equation.9.20}{}}
\newlabel{eq:14}{{21}{11}{Proof of Theorem \ref {thm:2}}{equation.9.21}{}}
\newlabel{eqn:ind1}{{22}{11}{Proof of Theorem \ref {thm:2}}{equation.9.22}{}}
\newlabel{eqn:ind2}{{23}{12}{Proof of Theorem \ref {thm:2}}{equation.9.23}{}}
\newlabel{eq:some-new-1}{{24}{12}{Proof of Theorem \ref {thm:2}}{equation.9.24}{}}
\newlabel{eq:some-new-2}{{25}{12}{Proof of Theorem \ref {thm:2}}{equation.9.25}{}}
\newlabel{eq:1corec}{{26}{12}{Proof of Theorem \ref {thm:2}}{equation.9.26}{}}
\newlabel{eq:some-new-6}{{27}{12}{Proof of Theorem \ref {thm:2}}{equation.9.27}{}}
\newlabel{eq:some-new-60}{{28}{12}{Proof of Theorem \ref {thm:2}}{equation.9.28}{}}
\@writefile{toc}{\contentsline {section}{\numberline {10}Proof of Theorem \ref  {thm:2-strong}}{13}{section.10}}
\newlabel{sec:4}{{10}{13}{Proof of Theorem \ref {thm:2-strong}}{section.10}{}}
\newlabel{eq:1-strong}{{31}{13}{Proof of Theorem \ref {thm:2-strong}}{equation.10.31}{}}
\newlabel{eq:2-strong}{{32}{13}{Proof of Theorem \ref {thm:2-strong}}{equation.10.32}{}}
\newlabel{eq:3-strong}{{33}{13}{Proof of Theorem \ref {thm:2-strong}}{equation.10.33}{}}
\newlabel{equ:crucial}{{34}{13}{Proof of Theorem \ref {thm:2-strong}}{equation.10.34}{}}
\newlabel{eq:4-strong}{{35}{14}{Proof of Theorem \ref {thm:2-strong}}{equation.10.35}{}}
\newlabel{eq:5-strong}{{36}{14}{Proof of Theorem \ref {thm:2-strong}}{equation.10.36}{}}
\@writefile{toc}{\contentsline {section}{\numberline {11}Proof of Theorem \ref  {thm:3}}{14}{section.11}}
\newlabel{sec:5}{{11}{14}{Proof of Theorem \ref {thm:3}}{section.11}{}}
\newlabel{eq:15}{{37}{14}{Proof of Theorem \ref {thm:3}}{equation.11.37}{}}
\newlabel{eq:16}{{38}{14}{Proof of Theorem \ref {thm:3}}{equation.11.38}{}}
\newlabel{eq:17}{{39}{14}{Proof of Theorem \ref {thm:3}}{equation.11.39}{}}
\newlabel{eq:18}{{40}{14}{Proof of Theorem \ref {thm:3}}{equation.11.40}{}}
\newlabel{eq:19}{{41}{15}{Proof of Theorem \ref {thm:3}}{equation.11.41}{}}
\newlabel{eq:20}{{42}{15}{Proof of Theorem \ref {thm:3}}{equation.11.42}{}}
\newlabel{eq:21}{{43}{15}{Proof of Theorem \ref {thm:3}}{equation.11.43}{}}
\newlabel{eq:22}{{44}{15}{Proof of Theorem \ref {thm:3}}{equation.11.44}{}}
\@writefile{toc}{\contentsline {section}{\numberline {12}Proof of Theorem \ref  {thm:3-new}}{16}{section.12}}
\newlabel{sec:6}{{12}{16}{Proof of Theorem \ref {thm:3-new}}{section.12}{}}
\newlabel{eq:1*}{{45}{16}{Proof of Theorem \ref {thm:3-new}}{equation.12.45}{}}
\newlabel{eq:2*}{{46}{16}{Proof of Theorem \ref {thm:3-new}}{equation.12.46}{}}
\newlabel{eq:3*}{{47}{16}{Proof of Theorem \ref {thm:3-new}}{equation.12.47}{}}
\newlabel{eq:changes13}{{48}{16}{Proof of Theorem \ref {thm:3-new}}{equation.12.48}{}}
\newlabel{eq:4*}{{49}{16}{Proof of Theorem \ref {thm:3-new}}{equation.12.49}{}}
\newlabel{eq:4*bis}{{50}{16}{Proof of Theorem \ref {thm:3-new}}{equation.12.50}{}}
\newlabel{eq:lt1}{{51}{16}{Proof of Theorem \ref {thm:3-new}}{equation.12.51}{}}
\newlabel{eq:11*}{{52}{17}{Proof of Theorem \ref {thm:3-new}}{equation.12.52}{}}
\newlabel{eq:12*}{{53}{17}{Proof of Theorem \ref {thm:3-new}}{equation.12.53}{}}
\@writefile{toc}{\contentsline {section}{\numberline {13}Examples of ARH numbers}{17}{section.13}}
\newlabel{sec:7}{{13}{17}{Examples of ARH numbers}{section.13}{}}
\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces ARH numbers with multipliers 1, 2, 3, 4, 5, 7 and without zero digits.}}{17}{table.1}}
\newlabel{t:1}{{1}{17}{ARH numbers with multipliers 1, 2, 3, 4, 5, 7 and without zero digits}{table.1}{}}
\newlabel{q:9}{{42}{18}{}{theorem.42}{}}
\newlabel{q:10-bis}{{43}{18}{}{theorem.43}{}}
\@writefile{toc}{\contentsline {section}{\numberline {14}Examples of MRH numbers}{18}{section.14}}
\newlabel{sec:8}{{14}{18}{Examples of MRH numbers}{section.14}{}}
\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces MRH numbers with multipliers $1,2,3,4,5$ and without zero digits.}}{18}{table.2}}
\newlabel{t:2}{{2}{18}{MRH numbers with multipliers $1,2,3,4,5$ and without zero digits}{table.2}{}}
\newlabel{q:12}{{45}{18}{}{theorem.45}{}}
\newlabel{q:13-bis}{{46}{18}{}{theorem.46}{}}
\@writefile{lot}{\contentsline {table}{\numberline {3}{\ignorespaces MRH numbers with $1,2,3,4,5,6, 7, 8$ digits and no zero digits.}}{19}{table.3}}
\newlabel{t:3}{{3}{19}{MRH numbers with $1,2,3,4,5,6, 7, 8$ digits and no zero digits}{table.3}{}}
\bibcite{B1}{1}
\bibcite{B2}{2}
\bibcite{C}{3}
\bibcite{CK}{4}
\bibcite{D}{5}
\bibcite{KD}{6}
\bibcite{G}{7}
\@writefile{toc}{\contentsline {section}{\numberline {15}Conclusion}{20}{section.15}}
\newlabel{sec:9}{{15}{20}{Conclusion}{section.15}{}}
\@writefile{toc}{\contentsline {section}{\numberline {16}Acknowledgments}{20}{section.16}}
\bibcite{FILS}{8}
\bibcite{Hardy}{9}
\bibcite{online}{10}
\bibcite{N}{11}