I am not able to find an answer to the following question:
For which positive even integers $k$ is the integer $$p^2+k$$ prime, where $p$ is a prime number $\gt5$?
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$p^2 \equiv 1(\mod 3)$
$p^2+ k \not \equiv 0(\mod 3) \implies k $ is of form $3n$ or $3n+1$.
$p^2 \equiv 1(\mod 2) \implies k \equiv 0( \mod 2)$ (since it is the stronger condition)
You can get $k$ in $(\mod 6)$,using CRT.
Similarily, you have $p^2 \equiv \pm 1\mod 5 \implies k \equiv 2,3$ or $0(\mod 5)$ . You can use CRT again to get $k \equiv x(\mod 30)$, you get a stronger condition.
Note that $\gcd(3,4)