Let us assume to the contrary and say that $\ell$ is a period. Let $k$ be a sufficiently large positive integer and define $m=999....9$ where we have $k$ nine's. Now, consider $a_m$. Clearly, $a_m=a_{m+\ell}$ and so $9k \equiv S(\ell -1)+1(\bmod 2)$ for all $k$. However, this fails since we have a contradiction for one of $k,k+1$ (they have different parities)

Hello, anantmudgal09. Why 9k≡l－1 mod 2? I think that this is wrong.Let S（n） denotes the sum of the digits of n.I think that 9k≡S（l－1）＋1 mod 2.