FAQFAQ   SearchSearch   MemberlistMemberlist   UsergroupsUsergroups 
 ProfileProfile   PreferencesPreferences   Log in to check your private messagesLog in to check your private messages   Log inLog in 
Forum index » Science and Technology » Math
Rational Points
Post new topic   Reply to topic Page 1 of 1 [10 Posts] View previous topic :: View next topic
Author Message
deepkdeb@yahoo.com
science forum beginner


Joined: 29 Dec 2005
Posts: 38

PostPosted: Fri Jul 07, 2006 1:40 am    Post subject: Rational Points Reply with 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.
Back to top
Rouben Rostamian
science forum addict


Joined: 01 May 2005
Posts: 85

PostPosted: Fri Jul 07, 2006 2:00 am    Post subject: Re: Rational Points Reply with quote

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?
Back to top
William Elliot
science forum Guru


Joined: 24 Mar 2005
Posts: 1906

PostPosted: Fri Jul 07, 2006 4:12 am    Post subject: Re: Rational Points Reply with quote

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.
Back to top
The World Wide Wade
science forum Guru


Joined: 24 Mar 2005
Posts: 790

PostPosted: Fri Jul 07, 2006 4:13 am    Post subject: Re: Rational Points Reply with quote

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.
Back to top
Robert B. Israel
science forum Guru


Joined: 24 Mar 2005
Posts: 2151

PostPosted: Fri Jul 07, 2006 6:25 am    Post subject: Re: Rational Points Reply with quote

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
Back to top
deepkdeb@yahoo.com
science forum beginner


Joined: 29 Dec 2005
Posts: 38

PostPosted: Fri Jul 07, 2006 12:36 pm    Post subject: Re: Rational Points Reply with quote

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.
Back to top
The Qurqirish Dragon
science forum Guru Wannabe


Joined: 30 Apr 2005
Posts: 104

PostPosted: Fri Jul 07, 2006 2:35 pm    Post subject: Re: Rational Points Reply with quote

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.
Back to top
Denis Feldmann2
science forum addict


Joined: 23 Apr 2006
Posts: 87

PostPosted: Fri Jul 07, 2006 3:58 pm    Post subject: Re: Rational Points Reply with quote

The Qurqirish Dragon a crit :
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.)



>
Back to top
OwlHoot
science forum beginner


Joined: 25 Jun 2006
Posts: 6

PostPosted: Sat Jul 08, 2006 1:27 pm    Post subject: Re: Rational Points Reply with quote

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 non-zero (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 non-zero 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 non-zero 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 crabby-looking "dot and carry one"
lines, but I'm typing in Google groups in a vile proportional-spacing
font
and Google has an extremely irritating habit of rearranging lines at
random!
Back to top
OwlHoot
science forum beginner


Joined: 25 Jun 2006
Posts: 6

PostPosted: Sat Jul 08, 2006 1:35 pm    Post subject: Re: Rational Points Reply with quote

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 non-zero (assumed hereafter) rational
solutions
if and only if there integer solutions. So it suffices to find
conditions for the
latter.
</pre>
</body>
</html>
Back to top
Google

Back to top
Display posts from previous:   
Post new topic   Reply to topic Page 1 of 1 [10 Posts] View previous topic :: View next topic
The time now is Wed Nov 14, 2018 10:36 am | All times are GMT
Forum index » Science and Technology » Math
Jump to:  

Similar Topics
Topic Author Forum Replies Last Post
No new posts Fixed points of a weakly continuous map f:H->H satisfying... Jack Stone Research 2 Mon Jul 10, 2006 3:12 pm
No new posts generalization of polynomial b-splines to rational martin2 Math 0 Mon Jul 10, 2006 7:36 am
No new posts Pi Is Rational CoreyWhite@gmail.com num-analysis 2 Tue Jul 04, 2006 7:25 pm
No new posts Linking points on algebraic curves with class numbers Dave Rusin Math 5 Wed Jun 28, 2006 8:30 pm
No new posts Why are there only finitely many Steiner points? tchow@lsa.umich.edu Research 7 Sat Jun 24, 2006 5:54 pm

Copyright © 2004-2005 DeniX Solutions SRL
Other DeniX Solutions sites: Electronics forum |  Medicine forum |  Unix/Linux blog |  Unix/Linux documentation |  Unix/Linux forums  |  send newsletters
 


Powered by phpBB © 2001, 2005 phpBB Group
[ Time: 0.0162s ][ Queries: 16 (0.0021s) ][ GZIP on - Debug on ]