Can this be solve using modular arithmetic? $k$ is prime $\Rightarrow$ $8k+1$ is prime Is the following statement true or false?

$\forall k  \in \mathbb{N}, k$ is prime $\Rightarrow$ $8k+1$ is prime

The answer is that the statement is false because if $k=7$, then $k$ is prime but $8k+1=57$ is not prime.
Is there are way to solve this problem using modular arithmetic? If yes, what is the process of solving it and how do I select the right modulo?

The answer is you can just use the first $k$ in your sequence of primes as modulus.
If for some prime $p$, the sequence of numbers $k_n$ defined by
$$k_n = \begin{cases}p, & n = 0\\8k_{n-1}+1, &n > 0\end{cases}$$
are all primes, we will have
$$k_{p-1} = 8^{p-1} p + 8^{p-2} + 8^{p-2} + \cdots 1 = 8^{p-1} p + \frac{8^{p-1}-1}{7}$$
is also a prime. If $p \ne 2$ nor $7$, then $\gcd(p,8) = 1$ and Fermat little theorem
leads to
$$p | 8^{p-1} - 1 \quad\implies\quad
p | \frac{8^{p-1}-1}{7} \quad\implies\quad 
p | k_{p-1}$$
This contradict with the assumption the sequence $k_n$ start at $k_0 = p$ are all primes.
Since we know the sequence $k_n$ starting with $k_0 = 2$ or $7$ are not all primes. We can conclude start with any prime $p$, the operation $k \mapsto 8k+1$ will fail to produce a prime after some finite step.
