Actually, the problem I'd really like to solve -- the one on positive definite quadratic forms was a special case of it -- is this. Instead of "positive definite quadratic forms in n variables" consider "polynomials in n variables which cannot produce negative output variables," and which input integers and output integers. Prove or disprove: there exists such a polynomial which assumes every natural number value (albeit perhaps we should permit a finite number of exceptions which cannot be represented) as output, and which represents each natural in at most a bounded number of ways. My theorem was a disproof in the case where the polynomial is quadratic. I suspect it can be disproven more generally, but if so, that is going to be harder... Pfister's solution of Hilbert's 17th problem is relevant to this.