Nice Equation 2

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Phlembac Adib Hasan
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Nice Equation 2

Unread post by Phlembac Adib Hasan » Thu Apr 12, 2012 4:33 pm

Find all primes $p$ such that $2^p+p^2$ is also a prime.(Really funny :lol: )
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Tahmid Hasan
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Re: Nice Equation 2

Unread post by Tahmid Hasan » Thu Apr 12, 2012 6:41 pm

Hint:
$\pmod 3$
,you can replace $2$ with a prime too and ask to find it in the problem.The result will be same.
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Phlembac Adib Hasan
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Re: Nice Equation 2

Unread post by Phlembac Adib Hasan » Thu Apr 12, 2012 6:48 pm

I think you saying this:
Hint :
$2\equiv -1(mod\; 3)$
And this solves the problem. :D
And I think replacing is not easy.If I replace $2$ by an odd prime, then the number will be even if $p$ is odd.So $p=2$.But can we say $p^2+4$ has finite solutions?
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Tahmid Hasan
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Re: Nice Equation 2

Unread post by Tahmid Hasan » Thu Apr 12, 2012 7:13 pm

Phlembac Adib Hasan wrote:I think you saying this:
Hint :
$2\equiv -1(mod\; 3)$
And this solves the problem. :D
And I think replacing is not easy.If I replace $2$ by an odd prime, then the number will be even if $p$ is odd.So $p=2$.But can we say $p^2+4$ has finite solutions?
Go on Adib,now use your favourite killing trick and get a contradiction. :)
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Re: Nice Equation 2

Unread post by photon » Sat Apr 14, 2012 12:47 pm

this was in the NT file of Masum bhai..simple..i made 2 problems similar.for $a,b$ prime show,
1)$a^b+b^a$ is prime
2)$a^a+b^b$ is prime

i can't solve the 2nd :( "perhaps" there are infinite prime in that 2nd form.
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Phlembac Adib Hasan
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Re: Nice Equation 2

Unread post by Phlembac Adib Hasan » Sat Apr 14, 2012 5:17 pm

It is an excalibur problem.So I am not surprised if you can find many sources.Because excalibur problems are a bit famous. :)
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