Given the equation $$x^2+2y^2+5z^2+xz =n$$ where $n$ is any positive integer, what is the smallest odd integer for which no integer solution $(x,y,z)$ exists (i.e. $x,y,z$ are integers)?
I know that for $n=10$ that no integer solution exists, but locating an odd number $n$ that has no integer solution has proven to be quite difficult.
Computer approaches seem to indicate that the number of integer solutions for odd $n$ grows as $n \to \infty$. It seems to me that in fact there are no integer solutions for any positive odd $n$. Do we see any proofs as to impossibilty (or can we find an odd $n$ that has no integer solution?)
It is an open conjecture that the quadratic form $Q(x,y,z)=x^2+2y^2+5z^2+xz$ represents all odd positive integers, see Conjecture 1 here. The same conjecture holds for the quadratic forms $$ Q(x,y,z)=x^2 +3y^2 +6z^2 +xy+2yz, $$ and $$ Q(x,y,z)=x^2 +3y^2 +7z^2 +xy+xz. $$ There is at present no general algorithm for determining the integers represented by a positive-definite ternary quadratic form $Q$.