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Let's consider the example,

$$ 3 + 11 + 19 = 33 \\ 2 + 31 = 33 $$

we can see that there are cases where the sum of 3 primes be expressed as the sum of 2 primes. However, I couldn't find a case where a sum of 3 primes couldn't be expressed as a sum of 2 primes.

Is there such a conjecture already in the world of mathematics? Or is there a counter example?

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  • $\begingroup$ Doesn't one of the two primes have to be 2? Unless 2 was one of the 3 primes... $\endgroup$
    – futurebird
    Aug 6 '15 at 13:46
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Write any odd number $n$ such that $n-2$ is not prime, as a sum of three primes to find your example. Explicitly $23= 5+7+11$ is not the sum of two primes.

Note that for an odd number to be sum of two primes you need one of the primes to be even and thus equal to $2$. So the only odd numbers that are sum oof two primes are $p+2$ where $p$ is an odd prime.

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