Is this a theorem?



> 
> Given any integer $m$, there exists $N(m)$, such that for all $n>N(m)$, we have
>  $$n = a\_1+a\_2+\cdots+a\_m$$
>  where $\gcd(a\_i,a\_j) = 1$ with $a\_i>1$ for all $i,j>1$.
> 
> 
> 


For instance, it is a textbook exercise to show for small values of $m$; Any integer exceeding $6$ is a sum of two relatively prime numbers; Any integer exceeding $17$ is a sum of three relatively prime numbers and so on.



Is every positive integer (eventually) a sum of $m$ relatively prime integers for every $m$?
We may induct on $m$. Assume that for $m-1$ this is proved. Choose $a\_1$ as a prime number between $n/2$ and $n-N(m-1)$ and apply induction proposition for $n-a\_1$.


