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I'm solving problem and I need your help, I know that every even integer can be expressed as sum of two primes and every integer can be expressed as sum of three primes. (for all integers <= 2 * 10^9)

But I want to know is there a way to check can we express odd number as sum of two primes.

Thanks in advance.

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  • $\begingroup$ The words "twin prime" come to mind. $\endgroup$ – hardmath Nov 27 '16 at 20:19
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    $\begingroup$ Technically, we don't "know" that every integer can be expressed as a sum of two primes. Goldbach's conjecture is still open, perhaps the most gotta-be-true unproven conjecture. $\endgroup$ – Joffan Nov 27 '16 at 20:21
  • $\begingroup$ I needed for number lower than 2*(10^9) and Goldbach's conjecture is proven for numbers up to 4*(10^18) $\endgroup$ – someone123123 Nov 27 '16 at 20:24
  • $\begingroup$ @someone123123: You are still wrong to claim that "every even integer can be expressed as sum of two primes". $\endgroup$ – TonyK Nov 28 '16 at 11:55
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Sure; if $n-2$ is prime, then yes, otherwise not.

Adding two numbers to get an odd number requires that one of them is odd and the other even, but since there is only one even prime ($2$), the test is simple.

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To make the sum of two numbers odd, one of the numbers must be odd and the other even. There is only one even prime, so that limits you to sums of the form $2+p$. Thus the odd numbers that are the sum of two primes are exactly the ones that are two more than a prime. The first few are $$ 5, 7, 9, 13, 15, 19, 21, 25, 31, 33, 39, 43\ldots $$ Also note that it is not known whether every even number is the sum of two primes. Every single even number that has been checked has been verified to be the sum of two primes, but we don't know whether it is always true.

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Note that every prime is odd, with the exception of 2, and also note that an odd number plus an odd number yields an even number. So if we have an odd number ( a prime) and we want to generate another odd number through addition, we have to add an even number. This means that the only odd numbers expressible as a sum of two primes are numbers of the form p+2, where p is a prime number.

The first odd number greater than one that can't be written as a sum of two primes is 3.

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    $\begingroup$ The first odd number greater than one that can't be written as a sum of two primes is 3. But then the next one is 11. $\endgroup$ – Arthur Nov 27 '16 at 20:23
  • $\begingroup$ You're right, that was an oversight on my part. Edited. $\endgroup$ – Ethan Baker Nov 27 '16 at 20:27

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