Someone came to me with the following observation: If $2n=p+q$ then $pq=n^2-m^2$ for some value of $0<m<n$ (namely, $m=n-p$ given $p\le q$).
Now he claims that this is actually equivalent: that the claim "For every $n$ there exists $0<m<n$ such that $n^2-m^2$ is the product of two primes" is equivalent to Goldbach's conjecture.
- Is it true? I tried proving the nontrivial direction but got stuck.
- Is it well known? I tried looking for references and couldn't find any.
(I am trying to explain to him that this is a hard conjecture and trivial observations are probably not worth his time except for recreation).