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The question is

If $x$ and $y$ are prime numbers , which of the following can not be their sum? $5$,$9$,$13$,$16$ or $23$.

The answer is $23$.

How did they get this? As far as I can tell is that when prime numbers are added I am suppose to get an even value for example $5+7 = 12 $ or $7+7 = 14 $ or $7+11 = 18 $. Could anyone please tell me what I am missing here?

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    It's called Goldbach's conjecture. Lots of info about it on the web.2012-08-25

4 Answers 4

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An odd integer $\,n\,$ is the sum of two primes iff $\,n-2\,$ is a prime, since $\,2\,$ is the only even prime...and, of course, the sum of two odd integers is an even one.

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List the first few primes $2,3,5,7,11,13,17,19,23,\ldots$, and observe that $5=2+3$, $9=2+7$, $13=2+11$ and $16=5+11$.

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    This is the same logic, I used but the one accepted is much smarter but either way works.2012-08-28
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All of the odd answers must have 2 as one of things you are adding. Otherwise, adding two primes always ends up as an even number. You can easily get 2 and 3, 2 and 7, 2 and 11. Now when you do 2 and 23, you see that the other number to add is 21. 21 is not prime, so 23 is your answer .

Basically, the methodology is to to take all the odd numbers and subtract 2 from it. If the result is not prime, then that will be your answer. If all odd numbers can be produced by 2 + a prime, then you'll have to guess and check for the even numbers.

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    Duely noted. Now I will go look up the Goldbach conjecture.2012-08-28
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Since $2$ is a also a prime, the sum of two primes is not necessarily even.