TITLE: Prove that $10101\ldots01$ can't be a perfect square.
QUESTION [6 upvotes]: Prove that $10101\ldots01$, where the sequence $"01"$ is repeated $k$ times can't be a perfect square for any value of $k$.
All I managed to prove is that the general formula is $1+\frac {100}{99}(100^k-1)$ but I don't know how to continue.

REPLY [5 votes]: An odd square has to be $\equiv 1 \bmod 8$ because$(2m+1)^2=4m^2+4m+1=4m(m+1)+1$ where $m(m+1)$ is even.  Any number rendered in base $10$ is a multiple of $1000=125×8$ plus its last three digits, thus congruent with the last three digits $\bmod 8$.  But $101 \equiv 5 \bmod 8$, a contradiction.