Incredible prime 2!

Let the required prime be 'p'

So , p = a + b ..............(1)where , a , b are primes.

Also , p = c-d ................(2) , where , c , d are primes .

Adding (1) , (2) , we get

2p = a + b + c - d ...........(3)

so , 2 divides a + b + c - d.

Let m = a + b + c , n = d

So , 2 | m - n , Either both m , n are odd or both are even.

1)If m is odd , a + b + c is odd , which means ,

(i) 2 of a,b,c are even and other is odd , which is not possible as there is only one even prime (2) .

(ii) One of a , b , c is even . let a = 2.

2) If both are even then , d = 2 ,

(iii) All are even , which is not possible.

(iv) 2 of a,b,c are odd and the other is even , and we have let a = 2.

So , we conclude that a = d =2 .

Putting a = d = 2 in (3) , we get ,

2p = b + c.

p = (b+c) / 2.

The smallest possible case is b = 3 , c = 7.

so , p = 5

I know that the question asks for smallest prime which is very easy by trial and error , but i wanted to enjoy the problem ! this would be helpful if they ask something other than smallest prime.

simple , i guessed 2+3=5 and also 7-2=5 thats it

The only prime number that we can get by the addition of two prime numbers is 5 which is 3+2=5 as well as 7-2=5.

the only possible prime that will lead to a sum that is odd is that it must be a an odd prime and an even prime, so the even prime must be 2 and the only possible prime that would be its pair is 3 so the ans. is 5

2+3 = 5
7-2 = 5