# S.O.S. Mathematics CyberBoard

Your Resource for mathematics help on the web!
 It is currently Sun, 19 May 2013 13:50:31 UTC

 All times are UTC [ DST ]

 Page 1 of 1 [ 5 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: The Nagell-Lutz TheoremPosted: Thu, 23 Feb 2012 16:18:44 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Wed, 10 Jan 2007 19:16:15 UTC
Posts: 650
Location: England
My Supervisor wants me to give a talk on part of my project, and decided that the Nagell-Lutz theorem was the most interesting, yet easiest to convey, piece of reading I had done.
The problem is, I know that the theorem isn't an "iff" statement - having with (where D is the discriminant of f(x)), does not mean that where is a point of finite order. Alas, I'm having trouble finding such a point.
I want to use the curve , and I thought that the point would be a good candidate (as it satisfies all of the conditions of the converse of the theorem). However, trying to calculate 2P I keep getting , which is not on the curve (). I can't work out what I'm doing wrong - the formulae I'm using is correct because I've used it plenty of other times to duplicate points, and once I work out the kinks with this stupid point, it will clearly be contradiction to the converse (as P will satisfy all of the conditions, but have infinite order). So can someone tell me where I'm going wrong? Because this is starting to irritate me.
The formulae I'm using are:
where and here
and
, where

_________________
"It's never crowded along the extra mile"

Graduated, and done with maths forever

Top

 Post subject: Re: The Nagell-Lutz TheoremPosted: Thu, 23 Feb 2012 16:43:08 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6004
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
peccavi_2006 wrote:
My Supervisor wants me to give a talk on part of my project, and decided that the Nagell-Lutz theorem was the most interesting, yet easiest to convey, piece of reading I had done.
The problem is, I know that the theorem isn't an "iff" statement - having with (where D is the discriminant of f(x)), does not mean that where is a point of finite order. Alas, I'm having trouble finding such a point.
I want to use the curve , and I thought that the point would be a good candidate (as it satisfies all of the conditions of the converse of the theorem). However, trying to calculate 2P I keep getting , which is not on the curve (). I can't work out what I'm doing wrong - the formulae I'm using is correct because I've used it plenty of other times to duplicate points, and once I work out the kinks with this stupid point, it will clearly be contradiction to the converse (as P will satisfy all of the conditions, but have infinite order). So can someone tell me where I'm going wrong? Because this is starting to irritate me.
The formulae I'm using are:
where and here
and
, where

The point (8,9) does not lie on the curve (8^3+17=529, not 81).

Try the point instead, for example. Alternatively, use the curve and , which is possibly the simplest (counter)example.

_________________

Top

 Post subject: Re: The Nagell-Lutz TheoremPosted: Thu, 23 Feb 2012 18:21:44 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Wed, 10 Jan 2007 19:16:15 UTC
Posts: 650
Location: England
ugh, this is what happens when people spring talks on me - I lose my head completely.

I was going to use (-1,4), as at some point I duplicate it for some reason (I think just to show how the formulas work), but I can't use it as a contradiction to the converse of the theorem because 4 doesn't divide the discriminant

I will totally use the other curve and your suggestion for P - thanks outermeasure

Edit: Oh! I know why I thought (8,9) was on the curve - I had for the x coordinate...and then square-rooted both sides because algebra is just that hard >.<

_________________
"It's never crowded along the extra mile"

Graduated, and done with maths forever

Top

 Post subject: Re: The Nagell-Lutz TheoremPosted: Fri, 24 Feb 2012 05:15:32 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6004
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
peccavi_2006 wrote:
ugh, this is what happens when people spring talks on me - I lose my head completely.

I was going to use (-1,4), as at some point I duplicate it for some reason (I think just to show how the formulas work), but I can't use it as a contradiction to the converse of the theorem because 4 doesn't divide the discriminant

I will totally use the other curve and your suggestion for P - thanks outermeasure

Edit: Oh! I know why I thought (8,9) was on the curve - I had for the x coordinate...and then square-rooted both sides because algebra is just that hard >.<

Why? For an elliptic curve , the discriminant is , so obviously 4 divides the discriminant.

Hmm... maybe you are using the version with modular discriminant instead?

_________________

Top

 Post subject: Re: The Nagell-Lutz TheoremPosted: Fri, 24 Feb 2012 14:09:29 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Wed, 10 Jan 2007 19:16:15 UTC
Posts: 650
Location: England
yep you're totally right - I stupidly copied down the equation for the discriminant as

Oh well - no one will notice it on the poster, I'm sure...

_________________
"It's never crowded along the extra mile"

Graduated, and done with maths forever

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 5 posts ]

 All times are UTC [ DST ]

#### Who is online

Users browsing this forum: No registered users

 You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forum

Search for:
 Jump to:  Select a forum ------------------ High School and College Mathematics    Algebra    Geometry and Trigonometry    Calculus    Matrix Algebra    Differential Equations    Probability and Statistics    Proposed Problems Applications    Physics, Chemistry, Engineering, etc.    Computer Science    Math for Business and Economics Advanced Mathematics    Foundations    Algebra and Number Theory    Analysis and Topology    Applied Mathematics    Other Topics in Advanced Mathematics Other Topics    Administrator Announcements    Comments and Suggestions for S.O.S. Math    Posting Math Formulas with LaTeX    Miscellaneous