Find all primes x such that f(x) is also a prime number. There is a function
\begin{align*}
f(x)=x^3 + x^2 +11x +2 \\
\end{align*}
Find all prime $x$ such that $f(x)$ is also a prime number.
I found that this is satisfied with an x value of 3 then the function is equal to 71, so both are primes, but I am unsure how to find other values or prove that there are no other existing solutions. I tried to use modular arithmetic, but I did not go so far.

Modular arithmetic is indeed the solution here. But first, we need to recall the following fact about prime numbers:

All primes $>3$ are $\pm1\pmod6$.

It's not too hard to see why. Numbers that are $0,2,4\pmod6$ are even while $3\pmod6$ is a multiple of $3$, leaving $1$ and $5$.
Now basically if $p$ is a prime that is $1\pmod6$, then $f(p)\equiv f(1)\equiv3\pmod6$, which cannot possibly be a prime (also $f(p)>3$). Similarly, if $p$ is $5\pmod6$ then $f(p)$ is also $3\pmod6$, which cannot be prime as well, so the only possible $x$ is $3$.
