TITLE: A conjecture about primes
QUESTION [7 upvotes]: Conjecture:

For each prime $p$ there are an infinite number of primes $q$ such
  that $p+q$ is a perfect square.

I have done a lot of tests using Bigz and I believe that it's possible to prove.

I change from possible to prove to not possible to disprove.
Define for all primes $q$, 
$w_q(n)=|\{p\in\mathbb P|p<n\wedge \exists k\in\mathbb N:p+q=k^2\}|$.
Then I've observed that it seems that for all primes $q$ there exists a natural number $N$ and a $\alpha\in\mathbb R_+$ such that $m>N$ implies that $w_q(10^m)>\alpha2^m$. This would imply the conjecture and is maybe possible to disprove.

REPLY [3 votes]: This would imply that for every prime $p$ , there are infinitely many primes of the form $n^2-p$ which is a well-known (open) problem.
It is a well known conjecture that if $m$ is not a perfect square, then there are infinitely many primes in $n^2-m$ , $n\in \mathbb{N}$ without saying anything if $m$ is prime.
Do not expect much.