Classically Integral Quadratic Forms Excepting at Most Two Values

• Published in: arXiv.org 2016-08
• Main Authors: Barowsky, Madeleine, Damron, William, Mejia, Andres, Saia, Frederick, Schock, Nolan, Thompson, Katherine
• Format: Article
• Language: English
• Subjects: Demand; Enumeration; Integers; Integrals; Quadratic forms; Theorems
• ISSN: 2331-8422

Let \(S \subseteq \mathbb{N}\) be finite. Is there a positive definite quadratic form that fails to represent only those elements in \(S\)? For \(S = \emptyset\), this was solved (for classically integral forms) by the \(15\)-Theorem of Conway-Schneeberger in the early 1990s and (for all integral forms) by the \(290\)-Theorem of Bhargava-Hanke in the mid-2000s. In 1938 Halmos attempted to list all weighted sums of four squares that failed to represent \(S=\{m\}\); of his \(88\) candidates, he could provide complete justifications for all but one. In the same spirit, we ask, "for which \(S = \{m, n\}\) does there exist a quadratic form excepting only the elements of \(S\)?" Extending the techniques of Bhargava and Hanke, we answer this question for quaternary forms. In the process, we prove what Halmos could not; namely, that \(x^2+2y^2+7z^2+13w^2\) represents all positive integers except \(5\). We develop new strategies to handle forms of higher dimensions, yielding an enumeration of and proofs for the \(73\) possible pairs that a classically integral positive definite quadratic form may except.