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
 Posts: 75
 Joined: Sun Mar 17, 2013 5:19 pm
 Location: Kushtia,Bangladesh.
 Contact:
Re: BdMO National 2013: Secondary 9, Higher Secondary 7
"The box said 'Requires Windows XP or better'. So I installed L$$i$$nux...:p"

 Posts: 2
 Joined: Mon Jan 13, 2014 6:00 pm
 Location: Kushtia,Bangladesh
 Contact:
Re: BdMO National 2013: Secondary 9, Higher Secondary 7
i p=2 then $2p+2q$. 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.