LuaLaTex - how to use number, computed later in the document

as in classic tex, you can use the aux file

\documentclass{article}


\gdef\mynumber{0}% first time
\begin{document}

The number will be \mynumber.

\makeatletter
\immediate\write\@auxout{\gdef\string\mynumber{\directlua{
n=1+2+3
tex.print(n)}}}
\makeatother

\end{document}

Welcome to TeX.SX! Here is a LaTeX3 implementation. It is independent of the TeX engine (LuaTeX, pdfTeX, XeTeX...). Moreover, as for LaTeX references, you will be warned if the result has changed since the last compilation run (meaning you should recompile to get correct output). The arithmetic expression passed to \writeresult uses the \int_eval:n function described in interface3.pdf (cf. part titled The l3int package).

\documentclass{article}
\usepackage{xparse}

\ExplSyntaxOn

\str_new:N \g_lemonbonbon_result_str
\str_new:N \g_lemonbonbon_previous_result_str

\msg_new:nnn { lemonbonbon } { additional-compilation-run-may-be-needed }
  { The~computed~result~has~changed;~another~compilation~run~may~be~needed. }

\cs_new_protected:Npn \lemonbonbon_write_result_to_aux_file:n #1
  {
    \cs_set_eq:cN { lemonbonbon@set@result } \relax % will not expand

    \iow_now:cx { @auxout }
      { \use:c { lemonbonbon@set@result } { \int_eval:n {#1} } }
  }

\NewDocumentCommand \writeresult { m }
  {
    \lemonbonbon_write_result_to_aux_file:n {#1}
  }

\NewDocumentCommand \useresult { }
  {
    \str_use:N \g_lemonbonbon_result_str
  }

\AtBeginDocument
  {
    % Save the previous result (this is run after the .aux file has been read)
    \str_gset_eq:NN \g_lemonbonbon_previous_result_str \g_lemonbonbon_result_str
  }

% Command run when the .aux file is reread, after the \AtEndDocument hook.
\cs_new_protected:Npn \lemonbonbon_warn_if_result_changed:n #1
  {
    \str_if_eq:VnF \g_lemonbonbon_previous_result_str {#1}
      {
        \msg_warning:nn { lemonbonbon }
                        { additional-compilation-run-may-be-needed }
      }
  }

% Reading the .aux file at end of document triggers comparison with the
% previous result in order to warn the user if it has changed.
\AtEndDocument
  {
    \cs_gset_eq:cN { lemonbonbon@set@result }
                   \lemonbonbon_warn_if_result_changed:n
  }

\makeatletter
% Command used in the .aux file (uses LaTeX2e naming conventions)
\NewDocumentCommand \lemonbonbon@set@result { m }
  {
    \str_gset:Nn \g_lemonbonbon_result_str {#1}
  }
\makeatother

\ExplSyntaxOff

\begin{document}
  The result is \useresult.

  \writeresult{3+4*(5-3)}
\end{document}

Output after two compilation runs: