TITLE: When does $axy+byz+czx$ represent all integers?
QUESTION [27 upvotes]: For which $a,b,c$ does $axy+byz+czx$ represent all integers?
In a recent answer, I conjectured that this holds whenever $\gcd(a,b,c)=1$, and I hope someone will know. I also conjectured that $axy+byz+czx+dx+ey+fz$ represents all integers when $\gcd(a,b,c,d,e,f)=1$ and each variable appears non-trivially, though I'm less optomistic about finding prior results on that. Here are some results:

If $\gcd(a,b)=1$ then $axy+byz+czx$ represents all integers. [Proof: Find $r,s$ with $ar+bs=1$, then take $x = r$, $y = n - crs$, $z = s$.]
$6xy+10yz+15zx$, the first case not covered above, represents all integers up to 1000. Similarly $77xy+91yz+143zx$ represents all integers up to 100. [by exhaustive search]
If $\gcd(a,b,c)=1$ then $axy+byz+czx$ represents all integers mod $p^r$. [proved in the above link]

The literature on this is hard to search because these are not positive definite forms, and many apparently relevant papers only consider the positive definite case. For old results, the most relevant parts of Dickson's History of the Theory of Numbers (v. 2, p. 434; v. 3, p. 224) mention only the case of $xy+xz+yz=N$. Does anyone here know a general result or reference?

REPLY [4 votes]: I have figured out some things; it is much quicker, as far as computing, to find a way for the Hessian matrix of the ternary quadratic form, is to have it represent the (two by two) Hessian of the form $xy;$ this form, or its quadratic space, is often called The Hyperbolic Plane; see page 15 in Cassels. 
Once this is done, there is just the business of adding an appropriate third row to "rows" to get a nice result. The final quadratic form is
(y - 1250*z)*x + (-797*z*y - 5751*z^2)
$$ xy -797yz - 1250 zx - 5751 z^2, $$
which is universal because we can take $z = 0, y = 1,$ and $x$ equal to the target number. Oh, the beginning form was your
$$ 77yz + 91 zx + 143xy $$
$$
\left(
\begin{array}{ccc}
830 &-3486 &-2145 \\
 616& -2587 & -1592 \\
 -3& -5& 12 \\
\end{array}
\right)
\left(
\begin{array}{ccc}
0 &143 &91 \\
 143& 0&77 \\
 91&77 &0 \\
\end{array}
\right)
\left(
\begin{array}{ccc}
830 &616 &-3 \\
 -3486& -2587 &-5 \\
 -2145&-1592 &12 \\
\end{array}
\right) =
\left(
\begin{array}{ccc}
 0&1 &-1250 \\
 1&0 & -797 \\
 -1250&-797 &-11502 \\
\end{array}
\right)
$$
Note: it turns out to be quite easy, with explicit matrices, to take the form with the visible hyperbolic plane to the form $xy - (abc) z^2,$ that GH has already proved equivalent to the original $ayz+bzx+cxy.$ 
$$
\left(
\begin{array}{ccc}
830 &-3486 &-2145 \\
 616& -2587 & -1592 \\
 1431507& -6012097& -3699553 \\
\end{array}
\right)
\left(
\begin{array}{ccc}
0 &143 &91 \\
 143& 0&77 \\
 91&77 &0 \\
\end{array}
\right)
\left(
\begin{array}{ccc}
830 &616 & 1431507 \\
 -3486& -2587 &-6012097 \\
 -2145&-1592 & -3699553 \\
\end{array}
\right) =
\left(
\begin{array}{ccc}
 0&1 &0 \\
 1&0 & 0 \\
 0& 0 &-2004002 \\
\end{array}
\right)
$$
========================================================
parisize = 4000000, primelimit = 500000
? h = [ 0,143,91; 143,0,77; 91,77,0]
%1 = 
[  0 143 91]

[143   0 77]

[ 91  77  0]

? rows = [ 830, -3486, -2145; 616, -2587, -1592 ]
%2 = 
[830 -3486 -2145]

[616 -2587 -1592]

? columns = mattranspose(rows)
%3 = 
[  830   616]

[-3486 -2587]

[-2145 -1592]

? rows * h * columns
%4 = 
[0 1]

[1 0]

? 
? 
? 
? rows = [ 830, -3486, -2145; 616, -2587, -1592; -3,-5,12 ]
%5 = 
[830 -3486 -2145]

[616 -2587 -1592]

[ -3    -5    12]

? matdet(rows)
%6 = 1
? columns = mattranspose(rows)
%7 = 
[  830   616 -3]

[-3486 -2587 -5]

[-2145 -1592 12]

? rows * h * columns
%8 = 
[    0    1  -1250]

[    1    0   -797]

[-1250 -797 -11502]

? x
%9 = x
? y
%10 = y
? z
%11 = z
? g = rows * h * columns
%12 = 
[    0    1  -1250]

[    1    0   -797]

[-1250 -797 -11502]

? vec = [ x,y,z]
%13 = [x, y, z]
? vect = mattranspose(vec)
%14 = [x, y, z]~
? vec * g * vect / 2 
%15 = (y - 1250*z)*x + (-797*z*y - 5751*z^2)
? 

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