6. The number of positive integer solutions $(x, y)$ for the indeterminate equation $x^{2}-2 y^{2}=5$ is ().
(A) 0
(B) 2
(C) 4
(D) infinitely many
6. A.

If the equation has a positive integer solution $(x, y)$, note that a perfect square is congruent to 0 or 1 modulo 4, thus, $x$ is odd, $y$ is even.
Let $x=2 m+1, y=2 n$. Substituting in, we get
$$
m(m+1)-2 n^{2}=1 \text {. }
$$

Since $m(m+1)$ is even, $2 n^{2}$ is even, this leads to a contradiction.
Therefore, the original equation has no positive integer solutions.