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\title{The Fifteen Theorem, and Generalizations}
%\title{A short proof of the Fifteen Theorem for Universal Quadratic Forms}
\author{Manjul Bhargava %(and JC? and WS?)
\\ Department of Mathematics\\
Princeton University\\Princeton, NJ 08544}
\begin{document}
\maketitle
\noindent{\bf 1. Introduction. }
In 1993, Conway and Schneeberger announced the following remarkable result:
\begin{theorem}[``The Fifteen Theorem'']
If a positive-definite quadratic form having integer matrix represents
every positive integer up to 15 then it represents every positive
integer.
\end{theorem}
The original proof of this theorem was never published, perhaps
because several of the cases involved rather intricate arguments.
A sketch of this original proof was given by Schneeberger in
\cite{Schneeberger}.
The purpose of this paper is 1) to give a short and direct proof of
the Fifteen Theorem, and 2) to give a summary of our recent results
towards some surprising extensions of the Fifteen Theorem. We intend
to provide more detailed accounts of these results in \cite{Bhargava} and
\cite{Bhargava2} respectively.
We note that our proof of the Fifteen Theorem is in spirit much the
same as that of the original unpublished arguments of Conway and
Schneeberger; however, we are able to avoid their intricate
case-by-case analysis, thereby obtaining a significantly simplified
proof. These simplified arguments are then applicable in
more general situations, which we shall outline in the final section.
\vspace{.2in}
\noindent
{\bf 2. Preliminaries. }
%As usual, a clarification of the word ``integral'' is in order. Depending
%on context, there are two standard meanings of ``integral quadratic form''.
%The first is what we may call an {\it integer-valued} form, i.e., a form
%taking integer values at all integers. The second is that which is called
%more precisely an {\it integer-matrix form}, i.e., a form whose matrix
%entries are integers. The Fifteen Theorem addresses only those forms
%satisfying the more restrictive second definition.
%It is worth remarking, however, that Conway and Schneeberger have formulated
%an analogous conjecture (``The 290 Conjecture'') for integer-valued forms.
%This conjecture seems significantly more difficult to prove, and may be
%addressed more fully in a future article. For the purposes of the current
%paper,
%Hence by an ``integral form'', or more briefly, ``form'', we shall always
%mean a positive definite quadratic form having integer matrix entries.
%for the more general case of integer-valued forms, namely:
%
%\begin{conjecture}(\em The 290 Conjecture.)
%If a positive-definite quadratic form taking integer values represents
%every positive integer up to 290 then it represents every positive integer.
%\end{conjecture}
%This conjecture appears to be significantly more difficult to prove, and
%will be addressed more fully in a future article.
%The key ingredient in the proof of the Fifteen Theorem is the {\it
%escalation process} (cite?).
The Fifteen Theorem deals with quadratic forms which are
positive-definite and have integer matrix. As is well-known, there is
a natural bijection between classes of such forms and lattices having
integer inner products; precisely, a quadratic form $f$ can be
regarded as the inner product form for a corresponding lattice $L(f)$.
Hence we shall oscillate freely between the language
of forms and the language of lattices. For brevity, by a ``form'' we
shall always mean a positive-definite quadratic form having integer
matrix, and by a ``lattice'' we shall always mean a lattice having
integer inner products.
A form (or its corresponding lattice) is said to be {\it universal} if
it represents every positive integer. If a form $f$ happens not to be
universal, define the {\it truant} of $f$ (or of its corresponding
lattice $L(f)$) to be the smallest positive integer not represented by
$f$.
Important in the proof of the Fifteen Theorem is the notion of
``escalator lattice.'' An {\it escalation} of a nonuniversal lattice
$L$ is defined to be any lattice which is generated by $L$ and a
vector whose norm is equal to the truant of $L$. An {\it escalator lattice}
is a lattice which can be obtained as the result of a sequence of
successive escalations of the zero-dimensional lattice.
\vspace{.2in}
\noindent
{\bf 3. Small-dimensional Escalators.} The unique escalation of the
zero-dimensional lattice is the lattice generated by a single vector
of norm $1$. This lattice corresponds to the form $x^2$ (or, in
matrix form, $[\, 1 \, ]$)
which fails to represent the number $2$. Hence an
escalation of $[\, 1\, ]$ has inner product matrix of the form
\[\left[\begin{array}{cc}
1 & a \\
a & 2 \end{array}\right].\]
By the Cauchy-Schwartz inequality, $a^2\leq 2$, so $a$ equals either
$0$ or $\pm 1$. The choices $a=\pm 1$ lead to isometric lattices, so we
obtain only two nonisometric two-dimensional escalators, namely those
lattices having Minkowski-reduced Gram matrices
{\footnotesize
$\left[\begin{array}{cc}
1 & 0 \\
0 & 1 \end{array}\right]$
}
and
{\footnotesize
$\left[\begin{array}{cc}
1 & 0 \\
0 & 2 \end{array}\right]$
}.
If we escalate each of these two-dimensional escalators in the same
manner, we find that we obtain exactly 9 new nonisometric escalator
lattices, namely those having Minkowski-reduced Gram matrices
{\scriptsize
$\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \end{array}\right],$
$\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 2 \end{array}\right],$
$\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 3 \end{array}\right],$
$\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 2 \end{array}\right],$
$\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 3 \end{array}\right],$
$\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 2 & 1 \\
0 & 1 & 4 \end{array}\right],$
$\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 2 & 1 \\
0 & 1 & 4 \end{array}\right],$
$\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 2 & 1 \\
0 & 1 & 5 \end{array}\right],$
}and
{\scriptsize
$\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 5 \end{array}\right]$}.
Escalating now each of these nine three-dimensional escalators, we find
exactly 207 nonisomorphic four-dimensional escalator lattices. All
such lattices are of the form $[1]\oplus L$, and the 207 such lattices $L$
are listed in Table 3. (We use the customary shorthand
``$D$: $a$ $b$ $c$ $d$ $e$ $f$'' to represent the three-dimensional
lattice {\tiny
$\left[\hspace{-.03in}\begin{array}{ccc}
a & f/2 & e/2 \\
f/2 & b & d/2 \\
e/2 & d/2 & c \end{array}\hspace{-.03in}\right]$} of determinant $D$.)
When attempting to carry out the escalation process just once more,
however, we find that many of the 207 four-dimensional lattices do not
escalate (i.e., they are universal). For instance, one of the
four-dimensional escalators turns out to be the lattice corresponding
to the famous four squares form, $a^2+b^2+c^2+d^2$, which is classically
known to represent all integers. The question arises: how many of the
four-dimensional escalators are universal?
\vspace{.2in}
\noindent{\bf 4. Four-dimensional Escalators.}
In this section, we prove that in fact 201 of the 207 four-dimensional
escalator lattices are universal; that is to say, only 6 of the
four-dimensional escalators can be escalated once again.
The proof of universality of these 201 lattices proceeds as follows.
In each such four-dimensional lattice $L_4$, we locate a 3-dimensional
sublattice $L_3$ which is known to represent some large set of
integers. Typically, we simply choose $L_3$ to be unique in its genus;
%(in fact, $L_3$ can always be taken to be one of a certain set of 14
%three-dimensional lattices).
%as is well-known, a lattice which is unique in its genus
in that case, $L_3$ represents
all integers that it represents locally (i.e., over each
$p$-adic ring $\Z_p$). Armed with this knowledge of $L_3$, we then show
that the direct sum of $L_3$ with its orthogonal complement in $L_4$
represents all sufficiently large integers $n \geq N$. A check of
representability of $n$ for all $n1$, then $r=u/t^2$ is also
not represented by $L_4$, contradicting the minimality of $u$.
Therefore $e=0$, and we have $u\equiv 7$ (mod $8$).
Now if $m\not\equiv 0$, $3$ or $7$ (mod $8$), then clearly $u-m$ is
not of the form $2^e(8k+7)$. Similarly, if $m\equiv 3$ or $7$ (mod
$8$), then $u-4m$ cannot be of the form $2^e(8k+7)$. Thus if
$m\not\equiv 0$ (mod $8$), and given that $u\geq 4m$, then either $u-m$
or $u-4m$ is represented by $L_3$; that is, $u$ is represented by
$L_3\oplus [m]$ (a sublattice of $L_4$) for $u\geq 4m$. An
explicit calculation shows that $m$ never exceeds $28$, and a computer
check verifies that every escalation $L_4$ of $L_3$ represents all
integers less than $4 \times 28=112$. It follows that any escalator
$L_4$ arising from $L_3$, for which the value of $m$ is not a multiple
of 8, is universal.
Of course, the argument fails for those $L_4$ for which $m$ is a
multiple of 8. We call such an escalation ``exceptional''.
Fortunately, such exceptional escalations are few and far between,
and are easily handled. For instance, an explicit
calculation shows that only two escalations of $L_3=${\scriptsize
$\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 2 \end{array}\right]$} are
exceptional (while the other 24 are not); these exceptional cases are
listed in Table 2.1. As is also indicated in the table, although
these lattices did escape our initial attempt at proof, the
universality of these four-dimensional lattices $L_4$ is still not any
more difficult to prove; we simply change the sublattice $L_3$ from
the escalator lattice {\scriptsize $\left[\begin{array}{ccc} 1 & 0 & 0
\\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{array}\right]$} to the ones listed in
the table, and apply the same argument!
%f any, can be determined in the following manner. Since $L_3$ has
%truant $7$, the Gram matrix of an escalator $L_4$ of $L_3$ takes the
%form
%\[\left[\begin{array}{cccc} 1 & 0 & 0 & a\\ 0 & 2 & 0 & b\\ 0 & 0
%& 2 & c\\ a & b & c & 7 \end{array}\right],
%\]
%where by the Cauchy-Schwartz inequality, we must have $|a|\leq 2$,
%$|b|\leq 3$, and $|c|\leq 3$. (We note also that, by changing the
%signs of the $v_i$ if necessary, we may assume that $a,b,c$ are
%nonnegative.) If $b$ and $c$ are both even, the orthogonal complement
%of $L_3$ in $L_4$ is generated by the vector $v=v_4-av_1-bv_2-cv_3$;
%otherwise it is generated by the vector $v=2v_4-2av_1-2bv_2-2cv_3$.
%Thus $m=7-a^2-b^2/2-c^2/2$ in the first case, and
%$m=28-4a^2-2b^2-2c^2$ in the second. (This verifies our assertion that
%$m\leq 28$.) In the first case, $m\leq 7$, and we see that $m$ cannot
%be a multiple of 8. So we assume without loss of generality that at
%least one of $b,c$ is odd. An examination of the expression
%$m=28-4a^2-2b^2-2c^2$ modulo 8 then reveals that $a$ must be even and
%both $b$ and $c$ must be odd if $m$ is to be a multiple of 8. We
%arrive $ These five triples lead to just two nonisomorphic exceptional
%lattices. Thes are given in Table 2.1.
\vspace{.075in}
It turns out that all of the 3-dimensional escalator lattices listed in
Table 1, except for the one labeled (6),
%{\scriptsize $\left[\begin{array}{ccc}
% 1 & 0 & 0 \\
% 0 & 2 & 1 \\
% 0 & 1 & 4 \end{array}\right],$}
are unique in their genus, so the universality of their escalations
can be proved by essentially identical arguments, with just a few
exceptions. As for escalator (6),
%{\scriptsize
%$\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & 1 \\ 0 & 1 & 4
%\end{array}\right],$}
%Kaplansky~\cite{Kaplansky} has shown %by elementary methods
although not unique in its genus, it does represent
all numbers locally represented by it
except possibly those which are $7$ or $10$ (mod~12).
%For completeness we give a short alternate proof of this fact.
Indeed, this escalator contains the lattice
{\scriptsize $\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 4 & 2 \\
0 & 2 & 8 \end{array}\right],$}
which is unique in its genus, and the lattices
{\scriptsize $\left[\begin{array}{ccc}
2 & $-2$ & 2 \\
$-2$ & 5 & 2 \\
2 & 2 & 8 \end{array}\right]$} and
{\scriptsize $\left[\begin{array}{ccc}
3 & 0 & 0 \\
0 & 5 & 4 \\
0 & 4 & 5 \end{array}\right]$,}
which together form a genus; a local check shows that the first genus
represents all numbers locally represented by escalator (6) which are
not congruent to 2 or 3 (mod~4), while the second represents
all such numbers not congruent to 1 (mod~3). The desired conclusion follows.
%Thus escalator (6) represents all numbers locally
%represented by it which are not congruent to 7 or 10 (mod 12), as claimed.
(This fact has been independently proven by
Kaplansky \cite{Kaplansky} using different methods.)
Knowing this, we may now proceed with essentially the same arguments
on the escalations of $L_3=${\scriptsize $\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 2 & 1 \\
0 & 1 & 4 \end{array}\right].$}
The relevant portions of the proofs for all nonexceptional cases are
summarized in Table~1.
%Among the nine three-dimensional escalators, only the
%fourth (as we have already seen), sixth and seventh yield exceptional
%escalators; they give rise to precisely two, two, and four such
%respectively, and these exceptional escalators are listed in Table 2.
``Exceptional'' cases arise only for escalators (4) (as we have
already seen), (6), and (7). Two arise for escalator (4). Although
four arise for escalator (6), two of them turn out to be
nonexceptional escalations of (1) and (8) respectively, and hence have
already been handled. Similarly, two arise for escalator (7), but
one is a nonexceptional escalation of (9). Thus only five truly
exceptional four-dimensional escalators remain, and these are listed in
Table~2.
In these five exceptional cases, other 3-dimensional sublattices unique
in their genus are given for which essentially identical arguments
work in proving universality. Again, all the relevant information is
provided in Table 2.
%This can be seen either by examining all the escalators explicitly, or by
%noting that
\vspace{.2in}
\noindent
{\bf 5. Five-dimensional Escalators.} As mentioned earlier, there are 6
four-dimensional escalators which escalate again; these are
listed in the first column of Table 4. A rather large calculation
shows that these 6 four-dimensional lattices escalate to an additional
1630 five-dimensional escalators! With a bit of fear we may ask again
whether any of these five-dimensional escalators escalate.
Fortunately, the answer is no; all five-dimensional escalators are
universal.
The proof is much the same as the proof of universality of
the four-dimensional escalators, but easier.
%In fact, we do not
%have to look at the five-dimensional escalators at all. Instead, we
%simply try to prove the universality of the 6 four-dimensional
%nonuniversal escalators.
We simply observe that, for the 6 four-dimensional nonuniversal
escalators, all parts of the proof of universality outlined in the
second paragraph of Section 4 go through---except for the final check.
The final check then reveals that each of these 6 lattices represent
every positive integer {\it except for one single number $n$}.
Hence once a single vector of norm $n$ is inserted in such a
lattice, the lattice must automatically become universal. Therefore
all five-dimensional escalators are universal. A list of the 6
nonuniversal four-dimensional lattices, together with the single numbers
they fail to represent, is given in Table 4.
Since no five-dimensional escalator can be escalated, it follows that
there are only finitely many escalator lattices: 1 of dimension
zero, 1 of dimension one, 2 of dimension two, 9 of dimension three,
207 of dimension four, and 1630 of dimension five, for a total of
1850.
\vspace{.2in}
\noindent
{\bf 6. Remarks on the Fifteen Theorem. } It is now obvious that
\vspace{.15in}
\noindent
(i) {\it Any universal lattice $L$ contains a universal sublattice
of dimension at most five.}
\vspace{.15in}
\noindent
For we can construct an escalator sequence $0=L_0\subseteq L_1\subseteq
\ldots$ within $L$, and then from Sections~4 and 5, we see that
either $L_4$ or (when defined) $L_5$ gives a universal escalator
sublattice of $L$.
Our next remark includes the Fifteen Theorem.
\vspace{.15in}
\noindent
(ii) {\it If a positive-definite quadratic form having integer matrix
represents the nine critical numbers $1$, $2$, $3$, $5$, $6$, $7$,
$10$, $14$, and $15$, then it represents every positive integer.
\noindent
(Equivalently, the truant of any nonuniversal form must be one of
these nine numbers.)
}
\vspace{.15in}
\noindent
This is because examination of the proof shows that only these
numbers arise as truants of escalator lattices.
We note that Remark (ii) is the best possible statement of the Fifteen Theorem,
in the following sense.
\vspace{.15in}
\noindent
(iii) {\it If $t$ is any one of the above critical numbers, then there is a
quaternary diagonal form that fails to represent $t$, but represents
every other positive integer.}
\vspace{.15in}
\noindent
Nine such forms of minimal determinant are
$[2,2,3,4]$ with truant $1$, $[1,3,3,5]$ with truant $2$,
$[1,1,4,6]$ with truant $3$, $[1,2,6,6]$ with truant $5$,
$[1,1,3,7]$ with truant $6$, $[1,1,1,9]$ with truant $7$,
$[1,2,3,11]$ with truant $10$, $[1,1,2,15]$ with truant $14$,
and $[1,2,5,5]$ with truant $15$.
However, there is another slight strengthening of the Fifteen Theorem,
which shows that the number 15 is rather special:
\vspace{.15in}
\noindent
(iv) {\it If a positive-definite quadratic form having integer matrix
represents every number below 15, then it represents every number
above 15.}
\vspace{.125in}
\noindent
This is because there are only four escalator lattices having truant
15, and as was shown in Section~5, each of these four escalators
represents every number greater than 15.
Fifteen is the smallest number for which Remark (iv) holds. In fact:
\vspace{.15in}
\noindent
(v) {\it There are forms which miss infinitely many integers starting
from any of the eight critical numbers not equal to 15.}
\vspace{.125in}
\noindent
Indeed, in each case one may simply take an appropriate escalator
lattice of dimension one, two, or three.
%(Six is a special number
%too, but perhaps need not be mentioned here!) Does anything need to
%be said here at all?]
\vspace{.125in}
\noindent
(vi) {\it There are exactly 204 universal quaternary forms.}
\vspace{.125in}
\noindent
An upper bound for the discriminant of such a form is easily determined;
a systematic use of the Fifteen Theorem then yields the desired result.
We note that the enumeration of universal quaternary
forms %(before the age of electronic computation)
was previously carried out in the well-known work of
Willerding~\cite{Willerding},
who ``showed'' that there are exactly 178 universal quaternary forms.
However, a comparison with our tables shows that Willerding actually
missed 36 universal forms, listed one universal form twice, and listed
9 non-universal forms! A list of all $204$ quaternary universal forms
is given in Table~5; the three entries not appearing among the list of
escalators in Table~3 have been italicized.
\vspace{.2in}
\noindent
{\bf 7. Extensions of the Fifteen Theorem. } Combining the techniques
exhibited here with analytic estimates for the growth of Fourier coefficients
of modular forms, we may obtain a series of remarkable generalizations
of the Fifteen Theorem, which show that the Conway-Schneeberger Fifteen
Theorem is far from being just an isolated coincidence!
In \cite{Bhargava2}, we obtain the following general finiteness result:
\begin{theorem}\label{finite}
Let $S$ be any subset of the nonnegative integers. Then there is
a unique subset $T\subseteq S$ satisfying the following properties:
\vspace{.05in}
$($i$)$ A positive-definite integer-matrix quadratic form represents $S$
if and only if it represents $T$;
\vspace{.05in}
$($ii$)$ if $T'$ also satisfies (i), then $T\subseteq T'$; and
\vspace{.05in}
$($iii$)$ $T$ is finite.
\end{theorem}
In other words, for any subset $S$ of the natural numbers,
there exists a unique, minimal finite subset $T$ of $S$,
such that a quadratic form represents $S$ if and only if it represents
$T$.
Interestingly, Theorem~\ref{finite} is not effective in general. But
for many choices of $S$ that that have been of much classical interest
in number theory, we have been able to effectively determine
the corresponding set $T$; namely, we have
\begin{itemize}
\item
If $S$ is the set of all natural numbers, then
$T=\{1,2,3,5,6,7,10,14,15\}$. (This is the Conway-Schneeberger
Fifteen Theorem.)
\item
If $S$ is the set of odd natural numbers, then
$T = \{1,3,5,7,11,15,33\}$.
\item
If $S$ is the set of prime numbers, then $T=
\{\mbox{primes up to }\,\,47\}\cup\{67,73\}$. (this is my favorite\,\, :)\,\,)
\end{itemize}
For several additional examples
of correspondences
$S \leftrightarrow T$, see \cite{Bhargava2}.
\vspace{.045in}
We note, finally, that we have examples of specific sets $S$ for which
$T$ seems essentially indeterminable without significant
progress on the Ramanujan conjecture for weight $3/2$ modular forms;
thus the general effectivization of Theorem~\ref{finite} is
yet another concrete number-theoretic
motivation for the resolution of Ramanujan's deep conjectures in the
subject of modular forms.
\vspace{.225in}
\noindent
{\bf Acknowledgments.} I thank Professor Conway
for many enlightening discussions, and for helpful comments on an early
draft of this paper. I am grateful to Prof.\ Bumcrot and the entire
Mathematics Department at Hofstra University for a wonderful time at
the retirement conference in honor of Prof.\ Bumcrot, May 11, 2001.
\vspace{-.05in}
\begin{thebibliography}{2}
\bibitem{Bhargava}
M.\ Bhargava, On the Conway-Schneeberger Fifteen Theorem,
AMS Contemporary Mathematics (Dublin), to appear.
\bibitem{Bhargava2}
M.\ Bhargava, Finiteness theorems for quadratic forms, in progress.
\bibitem{Kaplansky}
I.\ Kaplansky, The first non-trivial genus of positive definite ternary
forms, {\it Math.\ Comp.} {\bf 64} (1995), no.\ 209, 341--345.
\bibitem{Schneeberger}
W.\ A.\ Schneeberger, {\it Arithmetic and Geometry of Integral Lattices},
Ph.D.\ Thesis, Princeton University, 1995.
\bibitem{Willerding}
M.\ Willerding, {\it Determination of all classes of positive
quaternary quadratic forms which represent all (positive) integers},
Ph.D. Thesis, Ohio State University, 1948.
\end{thebibliography}
\noindent
\begin{tabular}{cccllc}
Three-dimensional& {} &Represents nos. &{}&{} & Check\\
\underline{ escalator lattice }&\underline{Truant}
&\underline{not of the form}&\underline{If $m$ }&\underline{Subtract}&
\underline{up to} \vspace{.2in}\\
\hspace{-.1in}(1)\hspace{.125in}$\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \end{array}\right]$\hspace{.15in}
& 7 & $2^eu_7$ & $\not\equiv 0$ (mod 8)& $m$ or $4m$& 112
\vspace{-.175in}\\
{}&{}&{}&$\equiv 0$ (mod 8)&does not arise&{-} \vspace{.2in}\\
\hspace{-.1in}(2)\hspace{.125in}$\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 2 \end{array}\right]$\hspace{.15in}
& 14 & $2^du_7$ &$\not\equiv 0$ (mod 16)& $m$ or $4m$& 224
\vspace{-.175in}\\
{}&{}&{}&$\equiv 0$ (mod 16)&does not arise&{-} \vspace{.2in}\\
\hspace{-.1in}(3)\hspace{.125in}$\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 3 \end{array}\right]$\hspace{.15in}
& 6 & $3^du_{-}$ & $\not\equiv 0$ (mod 9)& $m$, $4m$, or $16m$& 864
\vspace{-.175in}\\
{}&{}&{}&$\equiv 0$ (mod 9)&does not arise&{-} \vspace{.2in}\\
\hspace{-.1in}(4)\hspace{.125in}$\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 2 \end{array}\right]$\hspace{.125in}
& 7 & $2^eu_7$ & $\not\equiv 0$ (mod 8)& $m$ or $4m$& 112
\vspace{-.175in}\\
{}&{}&{}&$\equiv 0$ (mod 8)&[See Table 2]&{-} \vspace{.2in}\\
\hspace{-.1in}(5)\hspace{.125in}$\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 3 \end{array}\right]$\hspace{.125in}
& 10 & $2^du_5$ & $\not\equiv 0$ (mod 16)& $m$ or $4m$& 1440
\vspace{-.175in}\\
{}&{}&{}&$\equiv 0$ (mod 16)&does not arise&{-} \vspace{.2in}\\
\hspace{-.1in}(6)\hspace{.125in}$\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 2 & 1 \\
0 & 1 & 4 \end{array}\right]$\hspace{.125in}
& 7 & $7^du_{-}$ or & $\not\equiv 0,3,9$ (mod 12)
& {} & {}
\vspace{-.185in}\\
{}&{}&$7,10$ (mod $12$)&\&$\not\equiv 0$ (mod 49)&$m$, $4m$, or $9m$& 3087
\vspace{.045in}\\
{}&{}&{}&$\equiv 0$ (mod 49)&does not arise&- \vspace{.045in}\\
{}&{}&{}&$\equiv 0,3,9$ (mod 12)&[See Table 2]&{-} \vspace{.2in}\\
\hspace{-.1in}(7)\hspace{.125in}$\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 4 \end{array}\right]$\hspace{.125in}
& 14 & $2^du_7$ & $\not\equiv 0$ (mod 16)& $m$ or $4m$& 224
\vspace{-.2in}\\
{}&{}&{}&$\equiv 0$ (mod 8)&[See Table 2]&{-} \vspace{.2in}\\
\hspace{-.1in}(8)\hspace{.125in}$\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 2 & 1 \\
0 & 1 & 5 \end{array}\right]$\hspace{.125in}
& 7 & $2^eu_7$ & $\not\equiv 0$ (mod 8)& $m$ or $4m$& 252
\vspace{-.2in}\\
{}&{}&{}&$\equiv 0$ (mod 8)&does not arise&{-} \vspace{.2in}\\
\hspace{-.1in}(9)\hspace{.125in}$\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 5 \end{array}\right]$\hspace{.125in}
& 10 & $5^du_{-}$ & $\not\equiv 0$ (mod 25)& $m$ or $4m$& 4000
\vspace{-.2in}\\
{}&{}&{}&$\equiv 0$ (mod 25)&does not arise&{-} \\
\end{tabular}
\vspace{.025in}
\center{{\bf Table 1. } Proof of universality of four-dimensional
escalators (nonexceptional cases) }
\noindent
\begin{tabular}{cccccc}
``Exceptional''& New unique in & Unrepresented &{}&{} & Check\\
\underline{ Lattice } & \underline{genus sublattice}
&\underline{ numbers }&\underline{ $m$ }&\underline{ Subtract }&
\underline{up to} \vspace{.225in}\\
$\left[\begin{array}{cccc}
1 & 0 & 0 & 2\\
0 & 2 & 0 & 1\\
0 & 0 & 2 & 1\\
2 & 1 & 1 & 7 \end{array}\right]$ &
$\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 2 & 1 \\
0 & 1 & 3 \end{array}\right]$
& $5^du_{+}$ & 40 & $m$ or $4m$& 160 \vspace{.125in}\\
$\left[\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 2 & 0 & 1\\
0 & 0 & 2 & 1\\
0 & 1 & 1 & 7 \end{array}\right]$ &
$\left[\begin{array}{ccc}
2 & 0 & 1 \\
0 & 2 & 1 \\
1 & 1 & 7 \end{array}\right]$
& \begin{array}{cc}$2^eu_1,2^eu_5,$\\$2^du_3,2^du_7,3^du_+$\end{array}
& 1 & $m$ & 14 \vspace{.125in} \\
$\left[\begin{array}{cccc}
1 & 0 & 0 & 1\\
0 & 2 & 1 & 0\\
0 & 1 & 4 & 3\\
1 & 0 & 3 & 7 \end{array}\right]$ &
$\left[\begin{array}{ccc}
2 & 1 & 1 \\
1 & 4 & 0 \\
1 & 0 & 4 \end{array}\right]$
& $2^du_7$ & 1 & $m$, $4m$, or $9m$ & 9 \vspace{.125in} \\
$\left[\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 2 & 1 & 1\\
0 & 1 & 4 & 0\\
0 & 1 & 0 & 7 \end{array}\right]^\ast$ &
$\left[\begin{array}{ccc}
2 & 0 & 0 \\
0 & 4 & 2 \\
0 & 2 & 10 \end{array}\right]$
& $2^du_{7}$ & 90 & $m$ or $4m$ & 504 \vspace{.125in} \\
%$\left[\begin{array}{cccc}
% 1 & 0 & 0 & 3\\
% 0 & 2 & 0 & 0\\
% 0 & 0 & 4 & 2\\
% 3 & 0 & 2 & 14 \end{array}\right]$ &
%$\left[\begin{array}{ccc}
% 1 & 0 & 0 \\
% 0 & 2 & 0 \\
% 0 & 0 & 5 \end{array}\right]$
% & $5^du_-$ & 80 & $m$ or $4m$ & 320 \vspace{.3in}\\
$\left[\begin{array}{cccc}
1 & 0 & 0 & 1\\
0 & 2 & 0 & 0\\
0 & 0 & 4 & 2\\
1 & 0 & 2 & 14 \end{array}\right]$ &
$\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 4 & 2 \\
0 & 2 & 13 \end{array}\right]$
& $2^du_5,2^eu_3$ & 2 & $m$ or $4m$ & 8 \\
\end{tabular}
\vspace{.05in}
\center{{\bf Table 2. } Proof of universality of four-dimensional
escalators (exceptional cases)}
\begin{tabular}{lllll}
1: 1 1 1 0 0 0
&
16: 2 3 3 2 0 0
&
30: 2 4 4 2 0 0
&
49: 2 3 9 2 2 0
&
72: 2 5 8 4 0 0
\vspace{-.008in}\\
2: 1 1 2 0 0 0
&
17: 1 2 9 2 0 0
&
31: 2 3 6 2 2 0
&
49: 2 4 7 0 0 2
&
74: 2 4 10 2 2 0
\vspace{-.008in}\\
3: 1 1 3 0 0 0
&
17: 1 3 6 2 0 0
&
31: 2 4 5 0 2 2
&
49: 2 5 6 0 2 2
&
76: 2 4 10 0 2 0
\vspace{-.008in}\\
3: 1 2 2 2 0 0
&
17: 2 3 4 0 2 2
&
32: 2 4 4 0 0 0
&
50: 2 4 7 2 2 0
&
77: 2 5 9 4 2 0
\vspace{-.008in}\\
4: 1 1 4 0 0 0
&
18: 1 2 9 0 0 0
&
32: 2 4 5 4 0 0
&
{\it 50: 2 5 5 0 0 0}
&
78: 2 4 10 2 0 0
\vspace{-.008in}\\
4: 1 2 2 0 0 0
&
18: 1 3 6 0 0 0
&
33: 2 3 6 0 2 0
&
51: 2 3 9 0 2 0
&
78: 2 5 8 2 0 0
\vspace{-.008in}\\
4: 2 2 2 2 2 0
&
18: 2 2 5 2 0 0
&
{\it 33: 2 4 5 2 0 2}
&
52: 2 3 9 2 0 0
&
80: 2 4 10 0 0 0
\vspace{-.008in}\\
5: 1 1 5 0 0 0
&
18: 2 3 3 0 0 0
&
34: 2 3 6 2 0 0
&
52: 2 5 6 2 0 2
&
80: 2 4 11 4 0 0
\vspace{-.008in}\\
5: 1 2 3 2 0 0
&
18: 2 3 4 2 0 2
&
34: 2 4 5 2 2 0
&
52: 2 5 6 4 0 0
&
80: 2 5 8 0 0 0
\vspace{-.008in}\\
6: 1 1 6 0 0 0
&
19: 1 2 10 2 0 0
&
34: 2 4 6 4 0 2
&
53: 2 5 6 2 2 0
&
82: 2 4 11 2 2 0
\vspace{-.008in}\\
6: 1 2 3 0 0 0
&
19: 2 3 4 2 2 0
&
35: 2 4 5 0 0 2
&
54: 2 3 9 0 0 0
&
82: 2 5 9 4 0 0
\vspace{-.008in}\\
6: 2 2 2 2 0 0
&
20: 1 2 10 0 0 0
&
36: 2 3 6 0 0 0
&
54: 2 4 7 2 0 0
&
{\it 83: 2 5 9 2 2 0}
\vspace{-.008in}\\
7: 1 1 7 0 0 0
&
20: 2 2 5 0 0 0
&
36: 2 4 5 0 2 0
&
54: 2 5 6 0 0 2
&
85: 2 5 9 0 2 0
\vspace{-.008in}\\
7: 1 2 4 2 0 0
&
20: 2 2 6 2 2 0
&
36: 2 4 6 4 2 0
&
54: 2 5 7 4 2 2
&
86: 2 4 11 2 0 0
\vspace{-.008in}\\
7: 2 2 3 2 0 2
&
20: 2 4 4 4 2 0
&
36: 2 5 5 4 2 2
&
55: 2 3 10 2 2 0
&
87: 2 5 10 4 2 0
\vspace{-.008in}\\
8: 1 2 4 0 0 0
&
{\it 21: 2 3 4 0 2 0}
&
37: 2 5 5 4 2 0
&
55: 2 5 6 0 2 0
&
88: 2 4 11 0 0 0
\vspace{-.008in}\\
8: 1 3 3 2 0 0
&
22: 1 2 11 0 0 0
&
38: 2 4 5 2 0 0
&
55: 2 5 7 4 0 2
&
88: 2 4 12 4 0 0
\vspace{-.008in}\\
8: 2 2 2 0 0 0
&
22: 2 2 6 2 0 0
&
38: 2 4 6 0 2 2
&
56: 2 4 7 0 0 0
&
88: 2 5 9 2 0 0
\vspace{-.008in}\\
8: 2 2 3 2 2 0
&
22: 2 3 4 2 0 0
&
39: 2 3 7 0 2 0
&
56: 2 4 8 4 0 0
&
90: 2 4 12 2 2 0
\vspace{-.008in}\\
9: 1 2 5 2 0 0
&
22: 2 3 5 0 2 2
&
40: 2 3 7 2 0 0
&
57: 2 3 10 0 2 0
&
90: 2 5 9 0 0 0
\vspace{-.008in}\\
9: 1 3 3 0 0 0
&
23: 1 2 12 2 0 0
&
40: 2 4 5 0 0 0
&
58: 2 3 10 2 0 0
&
92: 2 4 13 4 2 0
\vspace{-.008in}\\
9: 2 2 3 0 0 2
&
23: 2 3 5 2 0 2
&
40: 2 4 6 2 0 2
&
58: 2 4 8 2 2 0
&
92: 2 5 10 4 0 0
\vspace{-.008in}\\
10: 1 2 5 0 0 0
&
24: 1 2 12 0 0 0
&
40: 2 4 6 4 0 0
&
58: 2 5 6 2 0 0
&
93: 2 5 10 2 2 0
\vspace{-.008in}\\
10: 2 2 3 2 0 0
&
24: 2 2 6 0 0 0
&
41: 2 4 7 4 0 2
&
58: 2 5 7 0 2 2
&
94: 2 4 12 2 0 0
\vspace{-.008in}\\
10: 2 2 4 2 0 2
&
24: 2 2 7 2 2 0
&
42: 2 3 7 0 0 0
&
60: 2 3 10 0 0 0
&
95: 2 5 10 0 2 0
\vspace{-.008in}\\
11: 1 2 6 2 0 0
&
24: 2 3 4 0 0 0
&
42: 2 4 6 0 0 2
&
60: 2 4 9 4 2 0
&
96: 2 4 12 0 0 0
\vspace{-.008in}\\
11: 1 3 4 2 0 0
&
24: 2 4 4 0 2 2
&
42: 2 4 6 2 2 0
&
60: 2 5 6 0 0 0
&
96: 2 4 13 4 0 0
\vspace{-.008in}\\
12: 1 2 6 0 0 0
&
24: 2 4 4 4 0 0
&
42: 2 5 5 4 0 0
&
61: 2 5 7 2 0 2
&
98: 2 4 13 2 2 0
\vspace{-.008in}\\
12: 1 3 4 0 0 0
&
25: 1 2 13 2 0 0
&
43: 2 3 8 2 2 0
&
62: 2 4 8 2 0 0
&
98: 2 5 10 2 0 0
\vspace{-.008in}\\
12: 2 2 3 0 0 0
&
25: 2 3 5 2 2 0
&
{\it 43: 2 5 5 2 0 2}
&
62: 2 5 7 4 0 0
&
100: 2 4 13 0 2 0
\vspace{-.008in}\\
12: 2 2 4 0 0 2
&
26: 1 2 13 0 0 0
&
44: 2 4 6 0 2 0
&
63: 2 5 7 0 0 2
&
100: 2 4 14 4 2 0
\vspace{-.008in}\\
13: 2 2 5 2 0 2
&
26: 2 2 7 2 0 0
&
45: 2 4 7 0 2 2
&
63: 2 5 7 2 2 0
&
100: 2 5 10 0 0 0
\vspace{-.008in}\\
13: 2 3 3 2 2 0
&
26: 2 4 4 2 2 0
&
45: 2 5 5 0 2 0
&
64: 2 4 8 0 0 0
&
102: 2 4 13 2 0 0
\vspace{-.008in}\\
14: 1 2 7 0 0 0
&
27: 1 2 14 2 0 0
&
45: 2 5 6 4 2 2
&
66: 2 4 9 2 2 0
&
104: 2 4 13 0 0 0
\vspace{-.008in}\\
14: 1 3 5 2 0 0
&
27: 2 3 5 0 2 0
&
46: 2 3 8 2 0 0
&
{\it 67: 2 5 8 4 2 0}
&
104: 2 4 14 4 0 0
\vspace{-.008in}\\
14: 2 2 4 2 0 0
&
27: 2 4 5 4 0 2
&
46: 2 4 6 2 0 0
&
68: 2 4 9 0 2 0
&
106: 2 4 14 2 2 0
\vspace{-.008in}\\
15: 1 2 8 2 0 0
&
28: 1 2 14 0 0 0
&
46: 2 5 6 4 0 2
&
68: 2 4 10 4 2 0
&
108: 2 4 14 0 2 0
\vspace{-.008in}\\
15: 1 3 5 0 0 0
&
28: 2 2 7 0 0 0
&
47: 2 4 7 2 0 2
&
68: 2 5 7 2 0 0
&
110: 2 4 14 2 0 0
\vspace{-.008in}\\
15: 2 2 5 0 0 2
&
28: 2 3 5 2 0 0
&
47: 2 5 6 4 2 0
&
70: 2 4 9 2 0 0
&
112: 2 4 14 0 0 0
\vspace{-.008in}\\
15: 2 3 3 0 2 0
&
28: 2 4 4 0 2 0
&
48: 2 3 8 0 0 0
&
70: 2 5 7 0 0 0
&
{}
\vspace{-.008in}\\
16: 1 2 8 0 0 0
&
28: 2 4 5 4 2 0
&
48: 2 4 6 0 0 0
&
72: 2 4 9 0 0 0
&
{}
\vspace{-.008in}\\
16: 2 2 4 0 0 0
&
30: 2 3 5 0 0 0
&
48: 2 5 5 2 0 0
&
72: 2 4 10 4 0 0\,
&
{}
\end{tabular}
\vspace{-.07in}
\center{{\bf Table 3. }Ternary forms $L$ such that $[1]\oplus L$ is
an escalator.\\
\small{(Entries not appearing in Table 5 are italicized.)}}
\begin{tabular}{cc}
\underline{Nonuniversal four-dimensional escalator} { }&
{ }\underline{Unique number not represented} \vspace{.325in} \\
$\left[\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 2 & 0 & 1\\
0 & 0 & 3 & 0\\
0 & 1 & 0 & 4 \end{array}\right]$ & 10 \vspace{.225in}\\
$\left[\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 2 & 1 & 0\\
0 & 1 & 4 & 1\\
0 & 0 & 1 & 5 \end{array}\right]$ & 10 \vspace{.225in}\\
$\left[\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 2 & 0 & 0\\
0 & 0 & 5 & 1\\
0 & 0 & 1 & 5 \end{array}\right]$ & 15 \vspace{.225in}\\
$\left[\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 2 & 0 & 0\\
0 & 0 & 5 & 0\\
0 & 0 & 0 & 5 \end{array}\right]$ & 15 \vspace{.225in}\\
$\left[\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 2 & 0 & 1\\
0 & 0 & 5 & 2\\
0 & 1 & 2 & 8 \end{array}\right]$ & 15 \vspace{.25in}\\
$\left[\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 2 & 0 & 1\\
0 & 0 & 5 & 1\\
0 & 1 & 1 & 9 \end{array}\right]$ & 15 \\
\end{tabular}
\vspace{.35in}
\center{{\bf Table 4. } Nonuniversal four-dimensional escalator lattices}
\begin{tabular}{lllll}
1: 1 1 1 0 0 0
&
16: 1 2 8 0 0 0
&
28: 2 3 5 2 0 0
&
48: 2 3 8 0 0 0
&
72: 2 4 9 0 0 0
\vspace{-.008in}\\
2: 1 1 2 0 0 0
&
16: 2 2 4 0 0 0
&
28: 2 4 4 0 2 0
&
48: 2 4 6 0 0 0
&
72: 2 4 10 4 0 0
\vspace{-.008in}\\
3: 1 1 3 0 0 0
&
16: 2 3 3 2 0 0
&
28: 2 4 5 4 2 0
&
48: 2 5 5 2 0 0
&
72: 2 5 8 4 0 0
\vspace{-.008in}\\
3: 1 2 2 2 0 0
&
17: 1 2 9 2 0 0
&
30: 2 3 5 0 0 0
&
49: 2 3 9 2 2 0
&
74: 2 4 10 2 2 0
\vspace{-.008in}\\
4: 1 1 4 0 0 0
&
17: 1 3 6 2 0 0
&
30: 2 4 4 2 0 0
&
49: 2 4 7 0 0 2
&
76: 2 4 10 0 2 0
\vspace{-.008in}\\
4: 1 2 2 0 0 0
&
17: 2 3 4 0 2 2
&
31: 2 3 6 2 2 0
&
49: 2 5 6 0 2 2
&
77: 2 5 9 4 2 0
\vspace{-.008in}\\
4: 2 2 2 2 2 0
&
18: 1 2 9 0 0 0
&
31: 2 4 5 0 2 2
&
50: 2 4 7 2 2 0
&
78: 2 4 10 2 0 0
\vspace{-.008in}\\
5: 1 1 5 0 0 0
&
18: 1 3 6 0 0 0
&
32: 2 4 4 0 0 0
&
51: 2 3 9 0 2 0
&
78: 2 5 8 2 0 0
\vspace{-.008in}\\
5: 1 2 3 2 0 0
&
18: 2 2 5 2 0 0
&
32: 2 4 5 4 0 0
&
52: 2 3 9 2 0 0
&
80: 2 4 10 0 0 0
\vspace{-.008in}\\
6: 1 1 6 0 0 0
&
18: 2 3 3 0 0 0
&
33: 2 3 6 0 2 0
&
52: 2 5 6 2 0 2
&
80: 2 4 11 4 0 0
\vspace{-.008in}\\
6: 1 2 3 0 0 0
&
18: 2 3 4 2 0 2
&
34: 2 3 6 2 0 0
&
52: 2 5 6 4 0 0
&
80: 2 5 8 0 0 0
\vspace{-.008in}\\
6: 2 2 2 2 0 0
&
19: 1 2 10 2 0 0
&
34: 2 4 5 2 2 0
&
53: 2 5 6 2 2 0
&
82: 2 4 11 2 2 0
\vspace{-.008in}\\
7: 1 1 7 0 0 0
&
19: 2 3 4 2 2 0
&
34: 2 4 6 4 0 2
&
54: 2 3 9 0 0 0
&
82: 2 5 9 4 0 0
\vspace{-.008in}\\
7: 1 2 4 2 0 0
&
20: 1 2 10 0 0 0
&
35: 2 4 5 0 0 2
&
54: 2 4 7 2 0 0
&
85: 2 5 9 0 2 0
\vspace{-.008in}\\
7: 2 2 3 2 0 2
&
20: 2 2 5 0 0 0
&
36: 2 3 6 0 0 0
&
54: 2 5 6 0 0 2
&
86: 2 4 11 2 0 0
\vspace{-.008in}\\
8: 1 2 4 0 0 0
&
20: 2 2 6 2 2 0
&
36: 2 4 5 0 2 0
&
54: 2 5 7 4 2 2
&
87: 2 5 10 4 2 0
\vspace{-.008in}\\
8: 1 3 3 2 0 0
&
{\it 20: 2 3 4 0 0 2}
&
36: 2 4 6 4 2 0
&
55: 2 3 10 2 2 0
&
88: 2 4 11 0 0 0
\vspace{-.008in}\\
8: 2 2 2 0 0 0
&
20: 2 4 4 4 2 0
&
36: 2 5 5 4 2 2
&
55: 2 5 6 0 2 0
&
88: 2 4 12 4 0 0
\vspace{-.008in}\\
8: 2 2 3 2 2 0
&
22: 1 2 11 0 0 0
&
37: 2 5 5 4 2 0
&
55: 2 5 7 4 0 2
&
88: 2 5 9 2 0 0
\vspace{-.008in}\\
9: 1 2 5 2 0 0
&
22: 2 2 6 2 0 0
&
38: 2 4 5 2 0 0
&
56: 2 4 7 0 0 0
&
90: 2 4 12 2 2 0
\vspace{-.008in}\\
9: 1 3 3 0 0 0
&
22: 2 3 4 2 0 0
&
38: 2 4 6 0 2 2
&
56: 2 4 8 4 0 0
&
90: 2 5 9 0 0 0
\vspace{-.008in}\\
9: 2 2 3 0 0 2
&
22: 2 3 5 0 2 2
&
39: 2 3 7 0 2 0
&
57: 2 3 10 0 2 0
&
92: 2 4 13 4 2 0
\vspace{-.008in}\\
10: 1 2 5 0 0 0
&
23: 1 2 12 2 0 0
&
40: 2 3 7 2 0 0
&
58: 2 3 10 2 0 0
&
92: 2 5 10 4 0 0
\vspace{-.008in}\\
10: 2 2 3 2 0 0
&
23: 2 3 5 2 0 2
&
40: 2 4 5 0 0 0
&
58: 2 4 8 2 2 0
&
93: 2 5 10 2 2 0
\vspace{-.008in}\\
10: 2 2 4 2 0 2
&
24: 1 2 12 0 0 0
&
40: 2 4 6 2 0 2
&
58: 2 5 6 2 0 0
&
94: 2 4 12 2 0 0
\vspace{-.008in}\\
11: 1 2 6 2 0 0
&
24: 2 2 6 0 0 0
&
40: 2 4 6 4 0 0
&
58: 2 5 7 0 2 2
&
95: 2 5 10 0 2 0
\vspace{-.008in}\\
11: 1 3 4 2 0 0
&
24: 2 2 7 2 2 0
&
41: 2 4 7 4 0 2
&
60: 2 3 10 0 0 0
&
96: 2 4 12 0 0 0
\vspace{-.008in}\\
12: 1 2 6 0 0 0
&
24: 2 3 4 0 0 0
&
42: 2 3 7 0 0 0
&
60: 2 4 9 4 2 0
&
96: 2 4 13 4 0 0
\vspace{-.008in}\\
12: 1 3 4 0 0 0
&
24: 2 4 4 0 2 2
&
42: 2 4 6 0 0 2
&
60: 2 5 6 0 0 0
&
98: 2 4 13 2 2 0
\vspace{-.008in}\\
12: 2 2 3 0 0 0
&
24: 2 4 4 4 0 0
&
42: 2 4 6 2 2 0
&
61: 2 5 7 2 0 2
&
98: 2 5 10 2 0 0
\vspace{-.008in}\\
12: 2 2 4 0 0 2
&
25: 1 2 13 2 0 0
&
42: 2 5 5 4 0 0
&
62: 2 4 8 2 0 0
&
100: 2 4 13 0 2 0
\vspace{-.008in}\\
{\it 12: 2 3 3 0 2 2}
&
{\it 25: 2 3 5 0 0 2}
&
43: 2 3 8 2 2 0
&
62: 2 5 7 4 0 0
&
100: 2 4 14 4 2 0
\vspace{-.008in}\\
13: 2 2 5 2 0 2
&
25: 2 3 5 2 2 0
&
44: 2 4 6 0 2 0
&
63: 2 5 7 0 0 2
&
100: 2 5 10 0 0 0
\vspace{-.008in}\\
13: 2 3 3 2 2 0
&
26: 1 2 13 0 0 0
&
45: 2 4 7 0 2 2
&
63: 2 5 7 2 2 0
&
102: 2 4 13 2 0 0
\vspace{-.008in}\\
14: 1 2 7 0 0 0
&
26: 2 2 7 2 0 0
&
45: 2 5 5 0 2 0
&
64: 2 4 8 0 0 0
&
104: 2 4 13 0 0 0
\vspace{-.008in}\\
14: 1 3 5 2 0 0
&
26: 2 4 4 2 2 0
&
45: 2 5 6 4 2 2
&
66: 2 4 9 2 2 0
&
104: 2 4 14 4 0 0
\vspace{-.008in}\\
14: 2 2 4 2 0 0
&
27: 1 2 14 2 0 0
&
46: 2 3 8 2 0 0
&
68: 2 4 9 0 2 0
&
106: 2 4 14 2 2 0
\vspace{-.008in}\\
15: 1 2 8 2 0 0
&
27: 2 3 5 0 2 0
&
46: 2 4 6 2 0 0
&
68: 2 4 10 4 2 0
&
108: 2 4 14 0 2 0
\vspace{-.008in}\\
15: 1 3 5 0 0 0
&
27: 2 4 5 4 0 2
&
46: 2 5 6 4 0 2
&
68: 2 5 7 2 0 0
&
110: 2 4 14 2 0 0
\vspace{-.008in}\\
15: 2 2 5 0 0 2
&
28: 1 2 14 0 0 0
&
47: 2 4 7 2 0 2
&
70: 2 4 9 2 0 0
&
112: 2 4 14 0 0 0
\vspace{-.008in}\\
15: 2 3 3 0 2 0
&
28: 2 2 7 0 0 0
&
47: 2 5 6 4 2 0\,
&
70: 2 5 7 0 0 0
&
{}
\end{tabular}
\vspace{.1in}
\center{{\bf Table 5. }Ternary forms $L$ such that $[1]\oplus L$ is
universal.\\
\small{(Entries not appearing in Table 3 are italicized.)}}
\end{document}