Classically Integral Quadratic Forms Excepting at Most Two Values

Madeleine Barowsky; William Damron; Andres Mejia; Frederick Saia; Nolan Schock; Katherine Thompson

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arXiv · 1608.01656

Classically Integral Quadratic Forms Excepting at Most Two Values

Madeleine Barowsky·William Damron·Andres Mejia·Frederick Saia·Nolan Schock·Katherine Thompson

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Abstract

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.

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Madeleine Barowsky, William Damron, Andres Mejia, Frederick Saia, Nolan Schock, Katherine Thompson. 2016-08-04. Classically Integral Quadratic Forms Excepting at Most Two Values. https://arxiv.org/abs/1608.01656

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