# Prove that $[f(7) - f (2)]$ is not a prime number

Let $$f(x) = ax^3 + bx^2 + cx + d$$ be the polynomial with $$a\in \mathbb Z.$$

Suppose that $$f(5) - f(4) =2019.$$

Prove that $$[f(7) - f (2)]$$ is not a prime number.

Thanks all for help!

$$f(5)-f(4) = 61a+9b+c = 2019$$.
It is easy to find that $$f(7)-f(2) = 335a+45b+5c = 335a+5(2019-61a) = 5(2019+6a)$$. Note that $$3\mid 2019$$ and $$3\mid 6a$$, we have $$2019+6a\ne \pm 1$$.
• Since $a\in\mathbb Z,$ so $(2019 + 6a)\neq \pm 1.$ Indeed, if $2019 + 6a = \pm 1$ then $a\in\mathbb Q.$ – Success May 10 at 4:16