Classically integral quadratic forms excepting at most two values

• Published in: Proceedings of the American Mathematical Society 2018-09, Vol.146 (9), p.3661-3677
• Main Authors: BAROWSKY, MADELEINE, DAMRON, WILLIAM, MEJIA, ANDRES, SAIA, FREDERICK, SCHOCK, NOLAN, THOMPSON, KATHERINE
• Format: Article
• Language: English
• Subjects: A. ALGEBRA, NUMBER THEORY, AND COMBINATORICS
• ISSN: 0002-9939; 1088-6826
• DOI: 10.1090/proc/13891

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 provide a new proof of the original outstanding conjecture of Halmos, 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.