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#1 2005-12-04 00:22:31

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

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?


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#2 2005-12-04 01:10:05

RickyOswaldIOW
Member
Registered: 2005-11-18
Posts: 212

Re: Integer system

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

Last edited by rickyoswaldiow (2005-12-04 01:10:15)


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#3 2005-12-04 01:39:40

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

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 ≤ y² ≤ x² ≤ a²
0 ≤ b² ≤ x² ≤ a²

By using the fact that squares have to be positive, so x² and y² must be positive, and thus b² has to be less than or equal to x², since y² is at least 0, and at most, x².


"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..."

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#4 2005-12-04 03:19:54

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

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?


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#5 2005-12-04 08:39:40

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

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..."

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#6 2005-12-04 08:45:41

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,534

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:

b² (<-y²->) x² (<-y²->) a²


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

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#7 2005-12-04 17:29:08

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

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:
|a1²+a2²=a3²
|a2²+a3²=a4²
|...
|a{N-2}²+a{N-1}²=aN²
have integer solutions?
I'm sure the upper system hasn't general solution.
Why?

Last edited by krassi_holmz (2005-12-04 17:31:08)


IPBLE:  Increasing Performance By Lowering Expectations.

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