  We consider the problem of classifying all positive-definite integer-valued
quadratic forms that represent all positive odd integers. Kaplansky considered
this problem for ternary forms, giving a list of 23 candidates, and proving
that 19 of those represent all positive odds. (Jagy later dealt with a 20th
candidate.) Assuming that the remaining three forms represent all positive
odds, we prove that an arbitrary, positive-definite quadratic form represents
all positive odds if and only if it represents the odd numbers from 1 up to
451. This result is analogous to Bhargava and Hanke's celebrated 290-theorem.
In addition, we prove that these three remaining ternaries represent all
positive odd integers, assuming the generalized Riemann hypothesis.
  This result is made possible by a new analytic method for bounding the cusp
constants of integer-valued quaternary quadratic forms $Q$ with fundamental
discriminant. This method is based on the analytic properties of Rankin-Selberg
$L$-functions, and we use it to prove that if $Q$ is a quaternary form with
fundamental discriminant, the largest locally represented integer $n$ for which
$Q(\vec{x}) = n$ has no integer solutions is $O(D^{2 + \epsilon})$.
