Definition: $n$ is a psuedo-square if the legendre symbol $(\frac{n}{p}) = 0$ or $= 1$ for $p = 3, 5, 7, 11$.
I want to find the probability of $n$ being a square or psuedo-square. I know that any square will satisfy the property of being a psuedo-square so it remains to find the probability of being a psuedo-square.
By Euler's Criterion:
$n^{\frac{p-1}{2}} \equiv (\frac{n}{p}) \mod p \iff$ $n$ is a quadratic residue modulo $p$.
So consider the cases for $n$ to be a psuedo-sqaure. That is $(\frac{n}{p}) = 0$ or $1$ when $p = 3,5,7,11$ and we get the following:
$n \equiv 1 \mod 3$ or $3|n$
$n^2 \equiv 1 \mod 5$ or $5|n$
$n^3 \equiv 1 \mod 7$ or $7|n$
$n^5 \equiv 1 \mod 11$ or $11|n$
Is this the correct line of thinking? How can I proceed to solve these congruences and then to find the probability?