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deepkdeb@yahoo.com science forum beginner
Joined: 29 Dec 2005
Posts: 38

Posted: Fri Jul 07, 2006 1:40 am Post subject:
Rational Points



The equation (1) represents an ellipse under the given conditions.
ax^2 + by^2 = r^2 (1) where r, b, a are integers > 0
Assertion: (1) cannot be satisfied for rational values of x, y.
Any comment upon the correctness of the assertion will be appreciated. 

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Rouben Rostamian science forum addict
Joined: 01 May 2005
Posts: 85

Posted: Fri Jul 07, 2006 2:00 am Post subject:
Re: Rational Points



In article <1152236444.887809.296900@m79g2000cwm.googlegroups.com>,
Deep <deepkdeb@yahoo.com> wrote:
Quote:  The equation (1) represents an ellipse under the given conditions.
ax^2 + by^2 = r^2 (1) where r, b, a are integers > 0
Assertion: (1) cannot be satisfied for rational values of x, y.
Any comment upon the correctness of the assertion will be appreciated.

Homework? 

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William Elliot science forum Guru
Joined: 24 Mar 2005
Posts: 1906

Posted: Fri Jul 07, 2006 4:12 am Post subject:
Re: Rational Points



On Thu, 6 Jul 2006, Deep wrote:
Quote:  The equation (1) represents an ellipse under the given conditions.
ax^2 + by^2 = r^2 (1) where r, b, a are integers > 0
Assertion: (1) cannot be satisfied for rational values of x, y.
Any comment upon the correctness of the assertion will be appreciated.
Look for counter examples. 


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The World Wide Wade science forum Guru
Joined: 24 Mar 2005
Posts: 790

Posted: Fri Jul 07, 2006 4:13 am Post subject:
Re: Rational Points



In article
<1152236444.887809.296900@m79g2000cwm.googlegroups.com>,
"Deep" <deepkdeb@yahoo.com> wrote:
Quote:  The equation (1) represents an ellipse under the given conditions.
ax^2 + by^2 = r^2 (1) where r, b, a are integers > 0
Assertion: (1) cannot be satisfied for rational values of x, y.
Any comment upon the correctness of the assertion will be appreciated.

If a = b = 1 and r = 5, then x = 3, y = 4 gives a solution. 

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Robert B. Israel science forum Guru
Joined: 24 Mar 2005
Posts: 2151

Posted: Fri Jul 07, 2006 6:25 am Post subject:
Re: Rational Points



In article <1152236444.887809.296900@m79g2000cwm.googlegroups.com>,
Deep <deepkdeb@yahoo.com> wrote:
Quote:  The equation (1) represents an ellipse under the given conditions.
ax^2 + by^2 = r^2 (1) where r, b, a are integers > 0
Assertion: (1) cannot be satisfied for rational values of x, y.
Any comment upon the correctness of the assertion will be appreciated.

One of the most obviously false assertions you've made so far.
It's certainly false if a or b is a square. Thus if b = c^2, try
x = 2 r/(a+1), y = r (a  1)/(c (a+1)).
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada 

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deepkdeb@yahoo.com science forum beginner
Joined: 29 Dec 2005
Posts: 38

Posted: Fri Jul 07, 2006 12:36 pm Post subject:
Re: Rational Points



Robert Israel wrote:
Quote:  In article <1152236444.887809.296900@m79g2000cwm.googlegroups.com>,
Deep <deepkdeb@yahoo.com> wrote:
The equation (1) represents an ellipse under the given conditions.
ax^2 + by^2 = r^2 (1) where r, b, a are integers > 0
Assertion: (1) cannot be satisfied for rational values of x, y.
Any comment upon the correctness of the assertion will be appreciated.
One of the most obviously false assertions you've made so far.
It's certainly false if a or b is a square. Thus if b = c^2, try
x = 2 r/(a+1), y = r (a  1)/(c (a+1)).
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

Yes, you are right Professor Israel. Kindly note the corrections " none
of a or b is a perfect square"
Now kindly give a counter example. 

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The Qurqirish Dragon science forum Guru Wannabe
Joined: 30 Apr 2005
Posts: 104

Posted: Fri Jul 07, 2006 2:35 pm Post subject:
Re: Rational Points



Deep wrote:
Quote:  Robert Israel wrote:
In article <1152236444.887809.296900@m79g2000cwm.googlegroups.com>,
Deep <deepkdeb@yahoo.com> wrote:
The equation (1) represents an ellipse under the given conditions.
ax^2 + by^2 = r^2 (1) where r, b, a are integers > 0
Assertion: (1) cannot be satisfied for rational values of x, y.
Any comment upon the correctness of the assertion will be appreciated.
One of the most obviously false assertions you've made so far.
It's certainly false if a or b is a square. Thus if b = c^2, try
x = 2 r/(a+1), y = r (a  1)/(c (a+1)).
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Yes, you are right Professor Israel. Kindly note the corrections " none
of a or b is a perfect square"
Now kindly give a counter example.

If a+b=r^2, then x=y=1 is a solution. 

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Denis Feldmann2 science forum addict
Joined: 23 Apr 2006
Posts: 87

Posted: Fri Jul 07, 2006 3:58 pm Post subject:
Re: Rational Points



The Qurqirish Dragon a écrit :
Quote:  Deep wrote:
Robert Israel wrote:
In article <1152236444.887809.296900@m79g2000cwm.googlegroups.com>,
Deep <deepkdeb@yahoo.com> wrote:
The equation (1) represents an ellipse under the given conditions.
ax^2 + by^2 = r^2 (1) where r, b, a are integers > 0
Assertion: (1) cannot be satisfied for rational values of x, y.
Any comment upon the correctness of the assertion will be appreciated.
One of the most obviously false assertions you've made so far.
It's certainly false if a or b is a square. Thus if b = c^2, try
x = 2 r/(a+1), y = r (a  1)/(c (a+1)).
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Yes, you are right Professor Israel. Kindly note the corrections " none
of a or b is a perfect square"
Now kindly give a counter example.
If a+b=r^2, then x=y=1 is a solution.

for a=2, b= 7 , r=1, x=y=1/3 is a solution (and also x=3/5, y=1/5 ; etc.)
> 

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OwlHoot science forum beginner
Joined: 25 Jun 2006
Posts: 6

Posted: Sat Jul 08, 2006 1:27 pm Post subject:
Re: Rational Points



Deep wrote:
Quote: 
The equation (1) represents an ellipse under the given conditions.
ax^2 + by^2 = r^2 (1) where r, b, a are integers > 0
Assertion: (1) cannot be satisfied for rational values of x, y.
Any comment upon the correctness of the assertion will be appreciated.

It's wrong.
By homogeneity there are nonzero (assumed hereafter) rational
solutions
if and only if there integer solutions. So it suffices to find
conditions for the
latter.
We can assume a and b are squarefree (each a product of distinct
primes),
because if say c^2 divides b we can consider an equation of the same
form
a.x^2 + (b/c^2).(c.y)^2 = r^2.
Then denoting a, b = e.f, e.g resp where GCD(f, g) = 1, clearly e is
also squarefree and thus divides r. So it suffices to consider:
f.x^2 + g.y^2 = e.(r/e)^2
Similarly, noting that GCD(GCD(e,f), g) = 1 and GCD(GCD(e,g), f) = 1,
it can be seen that GCD(e, f) (also squarefree) divides y and GCD(e,g)
divides x, and upon dividing these out it turns out that integer
solutions
to the original imply and are implied by integer solutions of:
A.X^2 + B.Y^2 + C.R^2 = 0
in which the integers A, B, C are coprime (relatively prime in pairs).
It isn't hard to prove that when A, B, C are coprime, then the latter
has nonzero integer solutions if and only if:
* A, B, C are not all of the same sign (true in your case,
because C < 0 < A, B)
* B.C, C.A, A.B are quadratic residues mod A, B, C resp.
For example 3.x^2 + 5.y^2 = 7.z^2 has no nonzero integer
solutions because, with A, B, C = 3, 5, 7 resp, 5.7 is not a
quadratic residue mod 3 (because 5.7 == 2 mod 3, whereas
a square is == 0 or 1 mod 3).
Apologies if this post come out with crabbylooking "dot and carry one"
lines, but I'm typing in Google groups in a vile proportionalspacing
font
and Google has an extremely irritating habit of rearranging lines at
random! 

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OwlHoot science forum beginner
Joined: 25 Jun 2006
Posts: 6

Posted: Sat Jul 08, 2006 1:35 pm Post subject:
Re: Rational Points



Deep wrote:
Quote: 
The equation (1) represents an ellipse under the given conditions.
ax^2 + by^2 = r^2 (1) where r, b, a are integers > 0
Assertion: (1) cannot be satisfied for rational values of x, y.
Any comment upon the correctness of the assertion will be appreciated.

(test please ignore)
<html>
<body>
<pre>
By homogeneity there are nonzero (assumed hereafter) rational
solutions
if and only if there integer solutions. So it suffices to find
conditions for the
latter.
</pre>
</body>
</html> 

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Google


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