The $15$ - theorem and the $290$ - theorem give sufficient conditions for a positive definite quadratic form to be universal (every natural number is a possible value of the quadratic form)

But how can I verify whether a positive definite form that is NOT universal, is "almost universal" (every sufficient large natural number is a possible value) ?

In particular, I found out that $3$ quadratic terms, like $a^2+2b^2+3c^2$ are not enough to create a universal quadratic form.

Can we also show that such a quadratic form ($ra^2+sb^2+tc^2$ with integers $0\le r\le s\le t$) cannot be "almost universal" ?


there is a general procedure that you can do for a particular quadratic form described here, and especially section 6.

I'm not an expert in the field, but here is a sketch of the procedure:

for a positive definite quadratic form $Q$, and some $m \in \mathbb N$ write

$$r_Q(m):=\#\{\vec{x} \in \mathbb N | Q(\vec{x})=m \}.$$

From these, there is a kind of generating function approach, where we take

$$\sum_{m \geq 1} r_Q(m)e^{2 \pi i nz}$$

which is a modular form.

Using the general theory, one wants to bound this away from zero, and show the coefficient to be positive for sufficiently large $m$. This can indeed be done using difficult bounds, but it can be done with computer algebra systems. Note that if one accomplishes this, then you get a finite list of integers that may not be represented, but the rest will be.

if you want to find all possible quadratic forms missing $S \subset \mathbb N$, (and to prove that you indeed have all of them) there is a process known as escalation, which can also be found in the 290 paper.

I believe that there is a folkloric theorem, Manjul's master theorem that shows that one can always play this game for any $S \subset \mathbb N$, although of course the larger $S$ is, the more difficult such computations become.

  • $\begingroup$ I hope that this answers your question $1$. For $2$, it can be shown (to my knowledge), but I do not know of a reference on hand $\endgroup$ – Andres Mejia Jun 26 '18 at 13:51

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