Suppose that x, y, x−y, and x+y are all positive prime numbers. What is the sum of the four numbers?

Well, I just guessed some values and I got the answer.

x=5, y=2, x−y=3, x+y=7. All the numbers are prime and the answer is 17.

Suppose if the numbers were very big, I wouldn’t have got the answer.

Do you know any ways to find the answer?

**Answer**

Note that x>y, since x−y is positive. Since x and y are both prime, this means that x must be greater than 2 and therefore odd. If y were odd, x+y would be an even number greater than 2 and hence not prime. Thus, y must be even, i.e., y=2.

Now we want an odd prime x such that x−2 and x+2 are both prime. In other words, we want three consecutive odd numbers that are all prime. But one of x−2,x, and x+2 is divisible by 3, so in order to be a prime it must **be** 3. Clearly that one must be x−2, the smallest of the three numbers, and we have our unique solution: x=5 and y=2, and x+y+(x+y)+(x−y)=3x+y=17.

**Attribution***Source : Link , Question Author : chndn , Answer Author : Brian M. Scott*