Distinguishing x25519 public keys from random?
A valid point of an elliptic curve in Weierstrass form satisfies the equation
$$
y^2 = x^3 + ax + b\,.
$$
We can rewrite this as $y = \pm \sqrt{x^3 + ax + b}$, which has either 2 solutions when $x^3 + ax + b$ is a square, 1 solution when $x^3 + ax + b = 0$, or 0 solutions when $x^3 + ax + b$ is not a square.

In X25519, we only see the $x$ coordinate, but we can still make use of the above. The curve equation is 
$$
y^2 = x^3 + 486662x^2 + x \bmod 2^{255}-19\,,
$$
from which we have that given a valid public key $x$, $x^3 + 486662x^2 + x$ must have a square root modulo $2^{255}-19$. On the other hand, a random string will be a square only around $1/2$ of the time. By inspecting $n$ candidate public keys, we can improve our certainty to $1 - 1/2^n$.