Classically integral quadratic forms excepting at most two values
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- by Madeleine Barowsky, William Damron, Andres Mejia, Frederick Saia, Nolan Schock and Katherine Thompson PDF
- Proc. Amer. Math. Soc. 146 (2018), 3661-3677 Request permission
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 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.References
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Additional Information
- Madeleine Barowsky
- Affiliation: Department of Mathematics, Wellesley College, Wellesley, Massachusetts, 02481
- Email: mbarowsk@wellesley.edu
- William Damron
- Affiliation: Department of Mathematics and Computer Science, Davidson College, Davidson, North Carolina, 28035
- Email: widamron@davidson.edu
- Andres Mejia
- Affiliation: Department of Mathematics, Bard College, Annandale-On-Hudson, New York, 12504
- Email: am8248@bard.edu
- Frederick Saia
- Affiliation: Department of Mathematics, Tufts University, Medford, Massachusetts, 02155
- Email: Frederick.Saia@tufts.edu
- Nolan Schock
- Affiliation: Department of Mathematics, California Polytechnic State University, San Luis Obispo, California, 93407
- Email: schocknol@gmail.com
- Katherine Thompson
- Affiliation: Department of Mathematical Sciences, DePaul University, Chicago, Illinois, 60614
- Email: kthompson0721@gmail.com
- Received by editor(s): April 22, 2017
- Received by editor(s) in revised form: June 28, 2017
- Published electronically: June 1, 2018
- Communicated by: Kathrin Bringmann
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 3661-3677
- MSC (2010): Primary 11E25; Secondary 11E20, 11E45
- DOI: https://doi.org/10.1090/proc/13891
- MathSciNet review: 3825823