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#1 2005-12-04 23:22:31

krassi_holmz
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Integer system

Can anybody solve the system
|x^2+y^2=a^2
|x^2-y^2=b^2,
where x, y, a and b are integers?


IPBLE:  Increasing Performance By Lowering Expectations.

#2 2005-12-05 00:10:05

RickyOswaldIOW
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Re: Integer system

What is the | for?  Is it a mathematical symbol?

Last edited by rickyoswaldiow (2005-12-05 00:10:15)


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#3 2005-12-05 00:39:40

Ricky
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Re: Integer system

a, b, x, and y = 0 is a solution, though uninteresting.

Other than that, I can only come up with the restrictions:

0 ≤ y ≤ x ≤ a
0 ≤ b ≤ x ≤ a

By using the fact that squares have to be positive, so x and y must be positive, and thus b has to be less than or equal to x, since y is at least 0, and at most, x.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

#4 2005-12-05 02:19:54

krassi_holmz
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Re: Integer system

Let write the system as:
|x^2+y^2=a^2
|b^2+y^2=x^2
Thus we get 2 pythagorean triples (Sorry if the syntax is incorrect. I don't know English well)
x=u^2-v^2
y=2uv
a=u^2+v^2
and
b=w^2-z^2
y=2wz
x=w^2+z^2.
To solve the system is enough to solve:
|2uv=2wz
|u^2-v^2=w^2+z^2
<=>
|uv=wz
|u^2=w^2+z^2+v^2
what to do further?


IPBLE:  Increasing Performance By Lowering Expectations.

#5 2005-12-05 07:39:40

Ricky
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Re: Integer system

What you're basically looking for is two pythagorean triplets that go:

b, y, x, and then y, x, a.

I don't believe any such triplets exist, although I can't prove it.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

#6 2005-12-05 07:45:41

MathsIsFun
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Re: Integer system

rickyoswaldiow wrote:

What is the | for?  Is it a mathematical symbol?

Just being used to "group" the two equations together - more typographic than mathematic. If it was |x| that would mean absolute value.

BTW, visually I get this:

b (<-y->) x (<-y->) a


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

#7 2005-12-05 16:29:08

krassi_holmz
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Re: Integer system

More colouricaly, we search for 4 squares such the sum of first and second is equal to third and the sum of second and third is equal to fourth. So we can make a generalized question. Does the system:
|a1+a2=a3
|a2+a3=a4
|...
|a{N-2}+a{N-1}=aN
have integer solutions?
I'm sure the upper system hasn't general solution.
Why?

Last edited by krassi_holmz (2005-12-05 16:31:08)


IPBLE:  Increasing Performance By Lowering Expectations.

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