## BdMO National 2013: Secondary 9, Higher Secondary 7

### BdMO National 2013: Secondary 9, Higher Secondary 7

If there exists a prime number $p$ such that $p+2q$ is prime for all positive integer $q$ smaller than $p$, then $p$ is called an "awesome prime". Find the largest "awesome prime" and prove that it is indeed the largest such prime.

- Fatin Farhan
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### Re: BdMO National 2013: Secondary 9, Higher Secondary 7

*"The box said 'Requires Windows XP or better'. So I installed*

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### Re: BdMO National 2013: Secondary 9, Higher Secondary 7

i p=2 then $2|p+2q$. So p cannot be 2.

Let p be the largest awesome prime.

The least value of q for any p is 1. So, we can write-

$ p+2\equiv1,2 (mod 3) $

now suppose,

$ p+2\equiv1 (mod 3) $

so, $p+4\equiv0 (mod 3) $

so, $2q=4$

and so, $q=2$. which is not possible.

So, for the largest awesome prime-

$ p+2\equiv2 (mod 3) $

or, $ p+6\equiv0 (mod 3) $

or, $ p\equiv0 (mod 3)$

Now we can say, p is a prime and divisible by 3. So the only value of $p=3$

And it's the largest awesome prime.

Let p be the largest awesome prime.

The least value of q for any p is 1. So, we can write-

$ p+2\equiv1,2 (mod 3) $

now suppose,

$ p+2\equiv1 (mod 3) $

so, $p+4\equiv0 (mod 3) $

so, $2q=4$

and so, $q=2$. which is not possible.

So, for the largest awesome prime-

$ p+2\equiv2 (mod 3) $

or, $ p+6\equiv0 (mod 3) $

or, $ p\equiv0 (mod 3)$

Now we can say, p is a prime and divisible by 3. So the only value of $p=3$

And it's the largest awesome prime.