For how many $n$ is $2021^n$ + $2022^n$ + $2023^n$ + ... + $2029^n$ prime?

Question

For how many $n$ is $2021^n$ + $2022^n$ + $2023^n$ + ... + $2029^n$ prime?

My first thought is set x = 2021 so we can create:

$x^n$ + $(x+1)^n$ + $(x+2)^n$ + ... + $(x+8)^n$

And then we expand each term even though we don't know $n$?

So for example: $(x+1)^n$ = $x^n$ + $\dbinom{n}{1}$$x^{n-1}$ + $\dbinom{n}{2}$ $x^{n-2}$ + ... + $\dbinom{n}{n-1}$ $x^{n-(n-1)}$ + $\dbinom{n}{n}$ $x^{n-n}$

I understand this, but how do I go about showing that the original sum is prime?

I don't think some obvious pattern will arise when I've expanded every term that will show whether or not it is prime.

Answer 1

Note that $x^n$, $(x+3)^n$ and $(x+6)^n$ give the same remaider when divided by $3$. So, the sum $x^n+(x+3)^n+(x+6)^n$ must be divisible by $3$.

Now, the expression in the question can be written as $$(2021^n+2024^n+2027^n)+(2022^n+2025^n+2028^n)+(2023^n+2026^n+2029^n)$$ which proves that this sum is divisible by $3$.

So, none of those expressions are prime.