2024/032) What is the proof that 11 is the only prime number of the form $n^2 + 2$ where n is prime?

We have a prime number is 2 or 3 of of the form $6n\pm  1$ for $n \gt 0$ let us compute $n^2+2$

$2^2 + 2 = 6 = 2 * 3$ not a prime

$3^2+ 2  = 11$ is a prime

$(6n\pm  1)^2+ 2 = (36n^2 \pm 12 n + 1) + 2 =  (36n^2 \pm 12 n + 3)  = 3(12n^2 \pm 4 n + 1)$ which is not a prime

hence 11 is the only prime