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.
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.)