Can Quadratic Forms Map Integers to Integers?

In summary, the conversation discusses whether a quadratic form, expressed as P(x,y) = a x + b y + d xy, can map integers onto integers. By changing the basis, the form can be re-expressed as P'(u,v) = Au^2 + Bv^2 for rational A and B. However, the problem reduces to whether P'(u,v) can map rationals onto integers, and the conclusion is that it cannot. The conversation also touches on the possibility of using matrices of determinant 1 to solve the problem, and the "sums of two squares theorem" of Fermat and Girard, which implies that the form U^2+V^2 cannot represent the integer 3.
  • #1
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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.
 
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  • #2
If ##A## isn't an integer, then ##P'(1,0)=A## isn't an integer.
 
  • #3
martinbn said:
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
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
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
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
Gear300 said:
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
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.
 

1. Can quadratic forms map all integers to other integers?

No, quadratic forms can only map certain integers to other integers. This is because quadratic forms are mathematical expressions that involve variables raised to the second power, which can result in non-integer values.

2. What are the conditions for a quadratic form to map integers to integers?

In order for a quadratic form to map integers to integers, the coefficients of the variables in the expression must also be integers. Additionally, the discriminant (b²-4ac) must be a perfect square.

3. Are there any specific examples of quadratic forms that can map integers to integers?

Yes, there are certain quadratic forms that can map integers to integers. Some examples include x², 2x²+3xy+2y², and 4x²+4xy+4y².

4. Can quadratic forms map rational numbers to integers?

Yes, quadratic forms can map rational numbers to integers if the rational numbers can be expressed as a ratio of two integers. For example, if a quadratic form can map 1/2 to an integer, it can also map 3/6, 4/8, etc. to integers.

5. What are some real-world applications of quadratic forms mapping integers to integers?

Quadratic forms can be used in number theory to study the properties of integers. They can also be used in cryptography and coding theory to encrypt and decode messages. Additionally, they have applications in computer graphics and physics to model real-world phenomena.

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