Basically, describe every number in the form $10^n + 1$ with square divisors meaning at least one of it's divisors is a square. Of course, there's infinite, but give a general algorithm for finding them. In a more mathematical way of putting this:
$10^n + 1 = mp^2$
Describe every solution $(n, p)$ to the above equation such that $n$ and $m$ are integers while $p$ is a prime.
If you would like a starting point, there are obviously no solutions when $p = 2, 3, 5$ but when $p = 7$, $m$ is an integer for all $n \equiv 21 \mod 42$, and when $p = 11$, $m$ is an integer for all $n \equiv 11 \mod 22$. I want to see if there is a pattern between n's that satisfy it. However, considering I want this spanning over every prime, perhaps there won't be a nice pattern between the n's but rather the p's. Basically, explore the problem, and if you find some sort of pattern, anything, do not hesitate to post.