prime number

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Sidharth
Posts: 20
Joined: Sat Feb 08, 2014 11:29 pm
Location: Naogaon,Bangladesh

prime number

Unread post by Sidharth » Tue Feb 18, 2014 12:12 am

If both p and $p^2+8$ are prime then prove that $p^3+4$ will also be a prime.

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asif e elahi
Posts: 183
Joined: Mon Aug 05, 2013 12:36 pm
Location: Sylhet,Bangladesh

Re: prime number

Unread post by asif e elahi » Tue Feb 18, 2014 11:31 am

If $p=2$,Then $p^{2}+8=12$ is not a prime.If $p$ is odd,then $p^{2}+8 \equiv 1+8\equiv 9\equiv 0(mod 3)$. So $3$ divides $p^{2}+8$.So there is no prime $p$ such that,$p^{2}+8$ is a prime.

Sidharth
Posts: 20
Joined: Sat Feb 08, 2014 11:29 pm
Location: Naogaon,Bangladesh

Re: prime number

Unread post by Sidharth » Tue Feb 18, 2014 1:53 pm

let,p=3.Now find $p^2+8$ and $p^3+4$

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asif e elahi
Posts: 183
Joined: Mon Aug 05, 2013 12:36 pm
Location: Sylhet,Bangladesh

Re: prime number

Unread post by asif e elahi » Tue Feb 18, 2014 5:17 pm

oh,$3$ is the only solution.

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