Prime values of binomial

Question

Does there always exist $x$, such that $x>b$ $x>a$ and $a+bx^n$ is prime? Of course, $a$, $b$ are given relatively prime numbers. I know that is true for n=1 in general, and I understand that it is an unsolved problem whether there are infinitely many of such $x$, so I just ask about existence.

Answer 1

No. Consider $$27+ 8x^3 = (2 x+3) (4 x^2-6 x+9)$$

which is always composite for positive integral values of $x$.

In general, $a^n + b^nx^n$ is always factorizable for $n$ odd:

$$a^n + b^nx^n = (a +bx)(a^{n-1} - a^{n-2}bx + \cdots - ab^{n-2}x^{n-2} + b^{n-1}x^{n-1})$$