Classically integral quadratic forms excepting at most two values

Abstract

Let S ⊆ N S \subseteq \mathbb {N} be finite. Is there a positive definite quadratic form that fails to represent only those elements in S S ? For S = ∅ S = \emptyset , this was solved (for classically integral forms) by the 15 15 -Theorem of Conway-Schneeberger in the early 1990s and (for all integral forms) by the 290 290 -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 } S=\{m\} ; of his 88 88 candidates, he could provide complete justifications for all but one. In the same spirit, we ask, “For which S = { m , n } S = \{m, n\} does there exist a quadratic form excepting only the elements of S 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 + 2 y 2 + 7 z 2 + 13 w 2 x^2+2y^2+7z^2+13w^2 represents all positive integers except 5 5 . We develop new strategies to handle forms of higher dimensions, yielding an enumeration of and proofs for the 73 73 possible pairs that a classically integral positive definite quadratic form may except.