Prove that no number in this list is prime - Formatting a proof advice

Question

Question: Let $n \in \mathbb{Z}$ where $n \geq 2$, prove no number in the list: $$n! + 2, n! + 3,...,n! + n$$ is prime.

I have written my proof exactly as follows:

Proof: $P(n) = n! + n = n((n-1)! + 1)$. Therefore both $n$ and $((n-1)! + 1)$ are factors, and therefore no number $p(n),n \geq 2$ is prime.

Is a proof of this form acceptable? Have I missed anything?

In my understand a proof should simply look like:

Statement->Proof->Conclusion

Is this correct?

Answer 1

Your proof is not acceptable as stated: you need to point out that neither $n$ nor $(n-1)!+1$ are 1. (This is easy, of course; you stated $n>2$ so the first term is not one, and $(n-1)!>0$ so $(n-1)!+1>1$.) As mentioned in breeden's answer, you haven't proved the theorem, either. You need to show

$n!+k$ is not prime for $2 \leq k \leq n$.

The argument proceeds similarly to before... instead of factoring out an $n$, you need to factor out a $k$.

Answer 2

You only proved that one number in the list is not prime, in particular the last number $n! + n$. What about the other $n - 2$ numbers: $n! + 2, n! + 3, \dots, n! + (n - 1)$?

Though your proof that the $n! + n$ is not prime is acceptable. Following the Statement -> Proof -> Conclusion format is recommended, though what is actually required depends on your audience (in your case, your professor and you!).

Answer 3

If $ 2 \leq k \leq n $: $$ k < n!+k=k(\frac{n!}{k}+1) \implies k|(n!+k)$$