Can Quadratic Forms Map Integers to Integers?

[Gear300]

Alright, so this might be a stupid question, but nevertheless, I ask. I am to consider whether the quadratic form
$P(x,y) = a x + b y + d xy$
can map the integers onto the integers. So through a change of basis, I re-express this as
$P'(u,v) = Au^2 + Bv^2$
for rational A and B. $u,v$ can be made to cover integer x,y for rational values, so the problem reduces to whether or not P'(u,v) can map the rationals onto the integers. I say no.

 

[martinbn]

If $A$ isn't an integer, then $P'(1,0)=A$ isn't an integer.

 

[Gear300]

If $A$ isn't an integer, then $P'(1,0)=A$ isn't an integer.

What's the implication thereafter? I generally understand that $u,v$ can be made to cover the integer values of $x,y$ for rational values, but that it more generally maps the rationals onto rationals (I edited my post above to be more clear about this).

 

[martinbn]

May be I misunderstood your question. I thought that you were asking if all the values are integers. Are you asking if all the integers are values of the form?

 

[Gear300]

Right. The inquiry is ultimately for $P(x,y)$. So in considering $x = \alpha u + \beta v$ and $y = \gamma u + \delta v$, I can restrict the coefficients to rational values and let $\alpha \delta = - \beta \gamma$ so that $u,v$ covers the integer values of x,y for rational values.

 

[martinbn]

Well, you need to say more about $A$ and $B$. If they are both positive you'll never get negative integers. If you are looking at non negative integers take $A=\frac12$ and $B=0$, then $P'=\frac{u^2}2$ and you cannot get odd integers.

 

[fresh_42]

Alright, so this might be a stupid question, but nevertheless, I ask. I am to consider whether the quadratic form
$P(x,y) = a x + b y + d xy$
can map the integers onto the integers. So through a change of basis, I re-express this as
$P'(u,v) = Au^2 + Bv^2$
for rational A and B. $u,v$ can be made to cover integer x,y for rational values, so the problem reduces to whether or not P'(u,v) can map the rationals onto the integers. I say no.

I would write it in matrix form. My suspicion is that matrices of determinant 1 are the answer.

 

[mathwonk]

since you do not give any conditions on the numbers a,b,c, or A,B, i am not sure what you are asking. But are you familiar with the famous "sums of two squares theorem" of Fermat (and Girard)? It implies for instance that the form U^2+V^2 cannot represent the integer 3, no matter what rational values are assigned to U,V.