How can i convert the form \begin{equation} \label{eq:(3)} {\overline{a \ldots ab \ldots b}}_{(10)} = y^2, \end{equation} (Suppose that $ 1 \leq a \leq 9$ and $0\leq b \leq 9$ are two integers, not necessarily equal)
into this diophantine equation?
$$ 10^{m}\cdot a \frac{10^{n} - 1}{9} +b\cdot\frac{10^m-1}{9}= y^2 , $$
Suppose there are $M$ digits in the period $$\overline{aa \ldots abb \ldots b}_{(10)}=0.aa\ldots abb\ldots b\space\space aa\ldots abb\ldots b\space \space aa\ldots abb\ldots b.......\\\overline{aa \ldots ab \ldots b}_{(10)}={aa \ldots abb \ldots b}\left(\frac{1}{10^{M}}+\frac{1}{10^{2M}}+\ldots\right)=\frac{aa \ldots abb \ldots b}{10^{M}}\left(1+\frac{1}{10^{M}}+\frac{1}{10^{2M}}\ldots \right)=\frac{aa \ldots abb \ldots b}{10^{M}}\left(\frac{1}{1-\frac{1}{10^{M}}}\right)\\\frac{aa \ldots abb \ldots b}{10^{M}}\left(\frac{10^{M}}{10^{M}-1}\right)=\frac{(aa \ldots a)10^m+bb \ldots b}{99\ldots9}$$
Can you follow to finish? (notice here there are $m$ digits $b$)