There exists $x \in \mathbb{R}$ such that the number $f(x)=x^2 +5x +4$ is prime."

Question

There exists $x \in \mathbb{R}$ such that the number $f(x)=x^2 +5x +4$ is prime.

I can't understand where to start.

This is what I have so far:

Let P(x) be the statement "$x^2 + 5x +4$ is prime". Then we have $\exists x \in \mathbb{R}, P(x)$.

I built a table and I suspect that this is not true. Hence, I'm attempting to disprove this statement. Apart from doing a direct proof, I can try a contrapositive proof but I don't know how to start.

Appreciate all inputs.

Answer 1

The function goes to infinity as x goes to infinity. It is also continuous. Thus it assumes all sufficiently large real numbers. In particular it assumes all sufficiently large prime numbers.

Answer 2

$f$ is continuous, and since $f(-1)=0$ and $$\lim_{x\to\infty} f(x)=\infty,$$ you can show that for every $p>0$, there exists some $x\in\mathbb R$ such that $f(x)=p$. You don't even have to restrict yourself to primes, the function reaches every positive number!

Answer 3

You could solve the equation $$ x^2 + 5x + 4 = 3 $$