Hello,

- Any integer can be written in the form 3k,3k+1 or 3k+2, for some integer

- Among these three possibilities, only 3k+1 and 3k+2 can be primes and 3 is a prime.

Now if

p+2is prime if and only if k+1=1, that is k=0. Otherwise, it is a multiple of 3 and hence is not prime.

But p=3x0+1=1 is not a prime.

Hence no prime number in the form 3k+1 satisfy the condition.

Now if

p+4is prime if and only if k+2=1, that is k=-1.

This gives p=-3, p+2=-2, p+4=1.

These are obviously not prime numbers. (they have to be positive).

Now if , ,

They're all prime numbers.

And it's the only possibility for p.