Problem
NT-B1-M01-P010 Three Primes in an Arithmetic Progression
#10
★★☆☆☆ Level 2 of 5
Find all primes \(p\) such that \(p\), \(p+4\), and \(p+8\) are all prime.
Consider the residues of the three numbers modulo \(3\).
If \(p>3\), then \(p\not\equiv0\pmod3\). If \(p\equiv1\pmod3\), then \(p+8\equiv0\pmod3\); if \(p\equiv2\pmod3\), then \(p+4\equiv0\pmod3\). In both cases, one number is greater than \(3\) and divisible by \(3\), hence composite. Checking \(p=3\), we get \(3,7,11\), all prime. The answer is \(p=3\).
Shows that primes are often controlled by small moduli.