## Nice Equation 2

- Phlembac Adib Hasan
**Posts:**1016**Joined:**Tue Nov 22, 2011 7:49 pm**Location:**127.0.0.1-
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### Nice Equation 2

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

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- Tahmid Hasan
**Posts:**665**Joined:**Thu Dec 09, 2010 5:34 pm**Location:**Khulna,Bangladesh.

### Re: Nice Equation 2

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

বড় ভালবাসি তোমায়,মা

- Phlembac Adib Hasan
**Posts:**1016**Joined:**Tue Nov 22, 2011 7:49 pm**Location:**127.0.0.1-
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### Re: Nice Equation 2

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?

Hint :

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
**Posts:**665**Joined:**Thu Dec 09, 2010 5:34 pm**Location:**Khulna,Bangladesh.

### Re: Nice Equation 2

Go on Adib,now use your favourite killing trick and get a contradiction.Phlembac Adib Hasan wrote: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?

বড় ভালবাসি তোমায়,মা

### Re: Nice Equation 2

this was in the NT file of Masum bhai..simple..i made 2 problems similar.

i can't solve the 2nd "perhaps" there are infinite prime in that 2nd form.

**for $a,b$ prime show,**

1)$a^b+b^a$ is prime

2)$a^a+b^b$ is prime1)$a^b+b^a$ 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
**Posts:**1016**Joined:**Tue Nov 22, 2011 7:49 pm**Location:**127.0.0.1-
**Contact:**

### Re: Nice Equation 2

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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