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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 =)

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  • $\begingroup$ What is your definition of prime number? Especially, can they also be negative? (This would actually IMO be an uncommon terminology.) $\endgroup$ – quid Jan 27 '15 at 12:24
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$$9-(n-2)^2=[3-(n-2)][3+(n-2)]=(5-n)(1+n)$$

The necessary condition : at least one of $5-n,1+n$ is $\pm1$

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  • $\begingroup$ Why is that the necessary condition? (+1) $\endgroup$ – Amad27 Jan 27 '15 at 12:20
  • $\begingroup$ @Amad27, Think of the reverse condition. $\endgroup$ – lab bhattacharjee Jan 27 '15 at 12:21
  • $\begingroup$ Perhaps because then a factor will be $1$? $\endgroup$ – Amad27 Jan 27 '15 at 12:25
  • $\begingroup$ 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 $\pm1$. $\endgroup$ – user160738 Jan 27 '15 at 12:26

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