# For how many integers is this a prime number?

For how many integers $n$ is: $9 - (n-2)^2$ a prime number?

I want to try this using a rigorous definition of prime number/ actual problem rather than try-error?

Please only give hints, so I can do this on my own!

Thanks =)

• What is your definition of prime number? Especially, can they also be negative? (This would actually IMO be an uncommon terminology.) – quid Jan 27 '15 at 12:24

$$9-(n-2)^2=[3-(n-2)][3+(n-2)]=(5-n)(1+n)$$
The necessary condition : at least one of $5-n,1+n$ is $\pm1$
• Perhaps because then a factor will be $1$? – Amad27 Jan 27 '15 at 12:25
• Well you can see that the expression is product of two integers and it will be prime if and only if one of these integers equal $\pm1$ and the other is not $0$ or $\pm1$. – user160738 Jan 27 '15 at 12:26