\documentclass[12pt]{amsart}
\usepackage{amsmath}
\usepackage{mathtools}
\mathtoolsset{showonlyrefs=true,showmanualtags=true}
%\mathtoolsset{showonlyrefs=true} %This numbers only equations that are labeled and referred to
       %Use \mathtoolsset{showonlyrefs=true,showmanualtags=true} to show also manual tags
       %For issues with subequations, use \noeqref to silently reference the equations, see:
       %https://tex.stackexchange.com/questions/62037/mathtools-showonlyrefs-fails-with-subequations

%% \usepackage[notcite, notref]{showkeys}
%% \usepackage{refcheck}
\begin{document}

This is a test, this is the equation environment with a label and a reference.
\begin{equation}
\label{eq:testshowonly:1}
\int 
\end{equation}
we belive that \eqref{eq:testshowonly:1} is correct 

This is a test, this is the equation environment with a label and
without a reference.
\begin{equation}
\label{eq:testshowonly:2}
\sum
\end{equation}


\end{document}