Almost universal integer quadratic forms

Question

This question is inspired by the 15-theorem. For any nonnegative integer k, define a k-universal integer quadratic form to be a form that represents all but k positive integers. So, universal forms are 0-universal forms. My question is, for every nonnegative integer k, is there a k-universal integer quadratic form?

Answer 1

yeah, sure. The ternary form $x^2 + y^2 + 2 z^2$ represents all positive odd numbers; also all even numbers except $4^k (16n + 14).$ As a result, if we take an odd number $N,$ the form $$ x^2 + y^2 + 2 z^2 + N w^2 $$ represents all numbers $n \geq N.$

To get a target number of misses $k,$ find the $k$th positive number missed by $x^2 + y^2 + 2 z^2,$ call this missed number $M.$ We know that $M$ is even. Let $N=M+1,$ which is then odd.

here is the list of 102 "regular" $A x^2 + B y^2 + C z^2,$ with $1 \leq A \leq B \leq C,$ also $\gcd(A,B,C)=1,$ ...