Density of the set of numbers of the form $x^2+2y^2$

Question

Is there a natural number $k$ such that for all intervals (of positive numbers) of length $k$, there is at least one element of the form $x^2+2y^2,$ where $x$ and $y$ are integers?

Answer 1

Let $p$ be a prime congruent to $5$ or $7$ modulo $8$. Then $p$ can only divide $x^2+2y^2$ if $p\mid x$ and $p\mid y$, so that then $p^2\mid(x^2+2y^2)$. Therefore if $m\equiv p\pmod{p^2}$ then $m$ cannot be of the form $x^2+2y^2$.

Let $p_1,\ldots,p_k$ be $k$ distinct primes of this form (there are infinitely many by Dirichlet). By the Chinese remainder theorem there is some $z\in\Bbb N$ with $z\equiv p_j-j\pmod{p_j^2}$ for all $j$. Then $z+j$ cannot be an $x^2+ny^2$. Then none of $z+1,\ldots,z+k$ is an $x^2+ny^2$.