Dirichlet’s Theorem on Arithmetic Progressions

Question

it states that an+d is prime infinitely often for n, given gcd(a,b)=1. So is kp+1 prime infinitely often, given p is a prime and k a fixed integer?

certainly, k cannot be odd as for odd primes, kp+1 is div by 2. so do all even k work? tia

Answer 1

If you fix k and let p run through the primes, Dirichlet's theorem says nothing about your situation, since you don't have an arithmetic progression.