Sum of Squares

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Main Question or Discussion Point

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.
 
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Answers and Replies

  • #2
martinbn
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If ##A## isn't an integer, then ##P'(1,0)=A## isn't an integer.
 
  • #3
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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).
 
  • #4
martinbn
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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?
 
  • #5
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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.
 
  • #6
martinbn
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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.
 
  • #7
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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.
 
  • #8
mathwonk
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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.
 

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