x^2 = 4*y^3 + 1
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From: M.A.Fajjal on 10 Aug 2010 22:55 Is there any rational solution for x^2 = 4*y^3 + 1
From: William Elliot on 11 Aug 2010 03:29 On Wed, 11 Aug 2010, M.A.Fajjal wrote: > Is there any rational solution for > > x^2 = 4*y^3 + 1 > x = +-1, y = 0
From: quasi on 11 Aug 2010 04:32 On Wed, 11 Aug 2010 02:55:20 EDT, "M.A.Fajjal" <h2maf(a)yahoo.com> wrote: >Is there any rational solution for > >x^2 = 4*y^3 + 1 Well, of course you have the 2 trivial solutions (x,y) = (1,0) (x,y) = (-1,0) Other than those, I have no idea. quasi
From: Timothy Murphy on 11 Aug 2010 05:29 M.A.Fajjal wrote: > Is there any rational solution for > > x^2 = 4*y^3 + 1 1. Multiply by 16: (4x)^2 = (4y)^3 + 16 So you can consider the curve y^2 = x^3 + 16. (Elliptic curves are usually written in this way.) 2. Look up a table of elliptic curves over the rationals, eg <http://www.asahi-net.or.jp/~KC2H-MSM/ec/eca1/ec01rp.txt>. You will find that this curve has rank 0, ie the group on the curve is finite. 3. The Nagell-Lutz theorem says that a point of finite order has integer coordinates. Also y^2 | 3D , where D is the discriminant, ie y^2 | 2^4 3^4 , or y | 36. 4. Going through the small number of possibilities, you will find that there are no non-trivial solutions.
From: achille on 11 Aug 2010 06:10
On Aug 11, 5:29 pm, Timothy Murphy <gayle...(a)eircom.net> wrote: > 3. The Nagell-Lutz theorem says that a point of finite order > has integer coordinates. > > Also y^2 | 3D , where D is the discriminant, ie > I'm confused, shouldn't it be y^2 | D instead of 3D ??? This is what's on wiki's entry and in an exercise of Silverman's book "Rational Points of Elliptic Curve".... |