Digit sum of a prime number [closed]

Question

Let 𝑝 be a positive integer and 𝑞 = 𝑆(𝑝) be the digit sum of 𝑝 such that 𝑞 + 1 ≡ 0 (mod 2). Is it that if 𝑝 is prime then 𝑞 is also prime?

e.g. 𝑝=47(prime)-> 𝑞=4+7=11 (prime)

Answer 1

Not necessarily, but it takes a while to see this, because you're not allowing $q$ to be a multiple of $2$, and it cannot be a multiple of $3$ (other than $3$ itself) because then the same would be true of $p$. So the smallest candidate for $q$ is $25$, which happens for the first time at $p=997$.