Because it must be $\displaystyle (a-b)(a+b)=101010$ then it must be $\displaystyle a+b=20202$.

So, we have to solve system of equations in order to see whether $\displaystyle a,b$ are integers.

$\displaystyle \begin{array}{l}

\left\{ \begin{array}{l}

a - b = 5 \\

a + b = 20202 \\

\end{array} \right. \\

\\

\left\{ \begin{array}{l}

a = 5 + b \\

a = 20202 - b \\

\end{array} \right. \\

\\

5 + b = 20202 - b \\

2b = 20202 - 5 \\

b = \frac{{20197}}{2} \\

\end{array}

$

Since $\displaystyle \frac{{20197}}{2}$ is not integer then b is not integer.

Is this proof valid?