If xx, yy, x+yx+y, and x−yx-y are prime numbers, what is their sum?

Suppose that x, y, xy, 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, xy=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 xy 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 x2 and x+2 are both prime. In other words, we want three consecutive odd numbers that are all prime. But one of x2,x, and x+2 is divisible by 3, so in order to be a prime it must be 3. Clearly that one must be x2, the smallest of the three numbers, and we have our unique solution: x=5 and y=2, and x+y+(x+y)+(xy)=3x+y=17.

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

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