Number Theory #5

Apart from 2, all the other primes are odd. Thus the difference/sum between any two primes other than 2 will be even.

Clearly 2 cannot be written as a sum of primes.

Now, we attempt to find a prime p > 2 which satisfies the given condition.

As mentioned earlier, p has to be the sum of a prime and 2 as well as the difference of a prime and 2, or else p is even and is not a prime.

$(p-2)$ , $p$ and $(p+2)$ are all primes. But one of them is divisible by 3. Thus, one of them has to be 3 in order for all of them to be prime. Checking, we find that only $(p-2)=3$ works.

$p=\boxed { 5 }$ is the only solution

5 = 2 + 3 = 7 - 2. Since all primes except 2 are odd, both the sum and difference of any two primes not including 2 will be even, and hence not prime. Thus 2 must be one of the two primes included in the sum and difference. So we require primes p,q > 2 such that p > q and p - 2 = q + 2 = r where r is prime. So in essence we need to find a prime number r such that (r - 2), r and (r + 2) are all primes. Clearly r = 5 satisfies this requirement, leading to the above answer. To prove that this is the only solution, Consider the following proof. For any 3 consecutive integers, one of then will be divisible by 3. So one of (r - 2), (r - 1) or r will be divisible by 3. With r = 5 we have (r - 2) = 3 divisible by 3, so now let r > 5. In this case, if either (r - 2) or r is divisible by 3 then they will not be prime, so assume that (r - 1) is divisible by 3. Then (r - 1) + 3 = r + 2 will also be divisible by 3 and hence not prime, (since r > 5). Thus there can be no r > 5 such that all of (r - 2), r and (r + 2) are all prime, and so my initial answer is the only solution to your question.

Let there be a no. N which is prime and which can be represented the way you ask.

All prime nos. are odd except 2. Let's keep 2 out of the equation for now.

Let, p1 + p2 = N and p4 - p3 = N

But p1, p2, p3, p4 are all odd as I am not considering 2 for now

And odd + odd = even

But N is not even as it itself is a prime

Similarly for p4 - p3 = N

This means any no. N which can be represented the way you say

should have one of the prime nos. as 2 in both p1 + p2

as well as p4 - p3. Only 5 is the prime which can be represented

that way