# Related to proving Goldbach's Conjecture

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 $P$

• One only needs to check this for even integers which are at most $2\cdot 3\cdot 5\cdot 7+49=259$, for then, if you have an even $N>259$, then you know that $N-210$ is a sum of two integers $a,b$ indivisible by $2,3,5,7$, and then $N=(a+210)+b$. Similarly, for any set of primes, we can reduce this to a finite computation. – Wojowu Nov 29 '16 at 21:02
• Wojowu, thanks for the comment. It is clearly not a significant result then. I did not use that method, so I did not realize that this result could be proven that easily. What do you think about the significance of the statement that every even number can be written as the sum of two primes or the difference of two numbers coprime to the set $P$? Like I said, I was not working toward proving these particular results, but I did not know if they are of value on their own. – limepickle Nov 29 '16 at 21:39