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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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2 is a prime.${}$ – Gerry Myerson Aug 25 '12 at 3:10
Thanks , so I could get an odd or an even result. Now am I suppose to try different combinations here of prime numbers ? Is there another way ? I mean I could go like $2+3 , 7+2 , 11+2 , ,13+3 $ – MistyD Aug 25 '12 at 3:14
I don't think there's another way. It is widely believed that every even number exceeding 2 is a sum of two primes, but no one has been able to prove this, so, given an even number, about all you can do is try different combinations of prime. Given an odd number. all you can do is subtract 2 and see if the result is prime. – Gerry Myerson Aug 25 '12 at 3:19
@GerryMyerson WOW. Thanks for the interesting fact!! – MistyD Aug 25 '12 at 3:25
It's called Goldbach's conjecture. Lots of info about it on the web. – Gerry Myerson Aug 25 '12 at 5:47
up vote 4 down vote accepted

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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Dear Sidd, Since the Goldbach conjecture has been checked up to some very large number, it is unlikely that a question like this would include an even number that is not the sum of two primes. So one can be pretty confident in practice that the answer will be odd. Regards, – Matt E Aug 28 '12 at 3:13
Duely noted. Now I will go look up the Goldbach conjecture. – mathguy Aug 28 '12 at 3:17

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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thats what I did. So I guess the only way is a hit and trial method – MistyD Aug 25 '12 at 3:17
No @Misty, is not the only way. Read below – DonAntonio Aug 25 '12 at 3:24
For some value of "below". – Gerry Myerson Aug 25 '12 at 5:48
This is the same logic, I used but the one accepted is much smarter but either way works. – Jeel Shah Aug 28 '12 at 3:45

Since $2$ is a also a prime, the sum of two primes is not necessarily even.

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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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