Problem
NT-B1-M01-P018 When \(p^2+2\) Is Prime
#18
★★★☆☆ Level 3 of 5
Find all primes \(p\) for which \(p^2+2\) is also prime.
Consider square residues modulo \(3\).
If \(p\ne3\), then \(p^2\equiv1\pmod3\), so \(p^2+2\equiv0\pmod3\). For \(p>3\), this number is greater than \(3\), hence composite. For \(p=2\), \(p^2+2=6\), also composite. For \(p=3\), we get \(11\), prime. The answer is \(p=3\).
A small but important olympiad check of the case \(p=2\).