<s>[INST] Infinitely many prime $n$ for which $n^2 = p + 8$ for some prime $p$. How to prove that there exist an infinite number of prime $n$ for which $n^2=p+8$ for some prime $p$?
Verification of the form $n^2=p+8$ where $n$ and $p$ are some $p$.
$$\begin{array}{|c|c|} \hline
       n     &         n^2 = 8 + p       \\
\hline
      11     &         121 = 8 + 113      \\
      23     &         529 = 8 + 521     \\
      31     &         961 = 8 + 953      \\
      37     &         1369= 8 + 1361    \\
\hline
\end{array}
$$
 [/INST] Your question is equivalent to there's infinitely many $m$ that:  
$2n^2 + 2(n - 2) = m,\ $where $2n + 1,\ 2m+1$ is prime.    
Such as: $n=3, \ 2n^2 + 2(n - 2) = 20,(2 \cdot 3+1)^2=7^2=49=2 \cdot 20+1+8=41+8,\ $ where $7,41$ is prime. It seems this is a elementary question.
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