From: M.A.Fajjal on
Is there any rational solution for

x^2 = 4*y^3 + 1
From: William Elliot on
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
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
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
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"....