Problem
NT-B1-M07-P023 Primes \(p\) and \(3^p+2\)
#23
★★★★☆ Level 4 of 5
Find all primes \(p\) such that \(p\mid 3^p+2\).
Modulo \(p\), \(3^p\) is congruent to \(3\).
By Fermat, \(3^p\equiv3\pmod p\) for every prime \(p\). Hence \(3^p+2\equiv5\pmod p\). If \(p\mid3^p+2\), then \(p\mid5\), so \(p=5\). Check: \(3^5+2=243+2=245\), divisible by \(5\). The answer is \(p=5\).
Similar to the previous one, but with a different base and constant.