How significant would it be to prove that every even number larger than $49$ can be written as the sum of two integers co-prime to $2, 3, 5,$ and $7$?
I came across this result incidentally when working on Goldbach's Conjecture. I was trying to prove that every even number less than $P_n^2$ could be written as the sum of two numbers co-prime to the set of consecutive primes $P = \{2, 3, ..., P_{n-1}\}$, which would in fact prove a result slightly stronger than Goldbach's Strong Conjecture, namely that every even number can be written as the sum of two primes greater than or equal to $P_n$. But working on this also gets you the result that $E > P_n^2$ can be written as the sum of two numbers co-prime to the set $P$, which is how I got the result in the first line. I'm having trouble getting past $7$, but perhaps someone (or eventually I) could figure out how to do it. Extending my proof would result in proving Goldbach's Conjecture, but I do not know if this result is significant enough to try and publish. From what I've seen, there hasn't been major progress in Goldbach's Strong Conjecture. I think that following my method would be easier than starting from scratch.
EDIT: It is also a proof showing that every even number can be written as the sum of two primes or the difference of two numbers coprime to the set $P$