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I was in the midst of proving a conjecture when I came across an observation that led me to forming a potentially new conjecture. The conjecture goes as follows:

Any given sum of twin primes (specifically the two primes in a twin prime pair, ie. $11$ and $13$) where the sum is greater than $12$, also represents the sum of 2 other distinct primes.

I've proven this for the first $1000$ twin primes and my computer is calculating beyond that set. Anyways, does anybody have any ideas of how I could go about proving this conjecture? I apologize if this conjecture has already been posed.

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    $\begingroup$ This OEIS sequence lists how many ways even integers can be written as sums of primes - and it suggests every even integer greater than $12$ is the sum of at least two distinct pairs of primes, which would imply your conjecture. Of course, we have no idea how to prove Goldbach's conjecture, and I would assume that your conjecture is not much easier. $\endgroup$
    – Milo Brandt
    Feb 15, 2015 at 1:17
  • $\begingroup$ Interesting, I noticed by hand calculation that it looked as if it applied to other numbers but I was hesitant to pose such as topic without examining it further. Thanks for the link! $\endgroup$
    – user1939991
    Feb 15, 2015 at 1:20
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    $\begingroup$ It's easy to come up with conjectures about prime numbers. But aye, proving them, that's the tough part. $\endgroup$
    – Mr. Brooks
    Feb 15, 2015 at 1:45
  • $\begingroup$ See my blog ideasfornumbertheory.wordpress.com for some possible insights. $\endgroup$
    – Sylvain Julien
    Feb 16, 2015 at 21:08
  • $\begingroup$ Also a special case of this mathSE conjecture. $\endgroup$
    – Caleb Stanford
    Jul 18, 2016 at 19:20

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There is a conjecture which states that the total number of writing $n$ as a sum of two odd primes is $\sim \frac{n}{2(\ln n)^2}$.
This shows that the sum of two twin primes will be a sum of two other primes as well.
So, your conjecture is a special case of an older conjecture.
(But ,I always liked elementary conjectures too,so do not be disappointed if your observation is "already stated" this has happened to me many times.)

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