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QUESTION

Use part (i) of question 4 above to prove that if $p$ is a prime
greater than $3$, then $p^2+2$ is composite.



ANSWER

If $p$ is a prime $>3$, then by question 4(i), either $p=6k+1$ or
$p=6k+5$ for some integer $k$. Thus $p^2+2$ is either
$(6k+1)^2+2=36k^2+12k+3$ or $(6k+1)^2+2=36k^2+60k+27$ and both of
these are divisible by 3. As $p^2+2>p>3$, $p^2+2$ must be
composite.





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