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This question already has an answer here:

I tried to solve this question but without a success: Let $p$ be a prime number,and $p^2+2$ is also prime, prove that $p=3$.

I tried to show $p^2+2$ as a product of numbers and then to show that $p=3$ is the only option that allows it to be prime. but I didn't find that presentation.

I would like to get help with this question, thanks

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marked as duplicate by lab bhattacharjee elementary-number-theory Apr 18 '15 at 13:23

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ Hint: All primes except for $2$ and $3$ are of the form $6n\pm1$. $\endgroup$ – Lucian Apr 18 '15 at 12:50
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if $p$ is not $3$ then $\gcd(p,3)=1$ so $3$ divides $(p-1)(p+1)$ hence $3$ divides $p^2-1+3=p^2+2$ so $p^2+2$ is not a prime

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