I am working on understanding Goldbach's conjecture and trying to make a small project on its various properties. Finally, I came up with the following statement,
"Let, $n>2$ be any natural number. Then there exist two prime numbers $p,q$ (not necessarily distinct) such that, $pq<n^2$ and $n^2-pq$ is a perfect square."
Can we prove it without assuming Goldbach's conjecture? Or is there any counterexample of my statement?
[Do not confuse with, Can you prove or disprove the following list of my conjectures?
For $n=3$ set $p=q=3$ we get $n^2-pq=0$ perfect square! [This case is special as here $n^2=pq$]
For $n=4$ set $p=5,q=3$ we get $n^2-pq=1$ perfect square!
For $n=5$ set $p=7,q=3$ we get $n^2-pq=4$ perfect square!
For $n=6$ set $p=5,q=7$ we get $n^2-pq=1$ perfect square! etc.
Any help would be highly appreciated. Thanks in advance!