Classically Integral Quadratic Forms Excepting at Most Two Values

TL;DR

This paper classifies quadratic forms that fail to represent at most two specific positive integers, extending previous theorems and providing explicit examples and enumeration for quaternary forms.

Contribution

It extends the classification of quadratic forms to those missing at most two values, solving a problem posed since Halmos and developing new methods for higher-dimensional forms.

Findings

Identified all 73 pairs of integers that can be excepted by such quadratic forms.

Constructed explicit quadratic forms for each exception pair.

Proved that the form x^2+2y^2+7z^2+13w^2 misses only the number 5.

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.