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Find all primes $p$ such that $2^p+p^2$ is also a prime.(Really funny )

Hint:,you can replace $2$ with a prime too and ask to find it in the problem.The result will be same.

I think you saying this:
Hint : And this solves the problem.
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?

Phlembac Adib Hasan wrote:I think you saying this:
Hint :

$2\equiv -1(mod\; 3)$

And this solves the problem.
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.

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.

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