Bill Allombert on Sun, 11 Aug 2013 22:39:08 +0200

 Re: x^3 + y^3 = 22 z^3 how to?

```On Sat, Aug 10, 2013 at 12:16:34AM +0200, Bill Allombert wrote:
> On Fri, Aug 09, 2013 at 03:20:35PM -0600, Elim Qiu wrote:
> > I'm finding positive int solution(s) (x,y,z) of the equation x^3 + y^3 = 22
> > z^3
> > other than the multiples of (17299,25469,9954).
> >
> > I did some code with iphthon but don't know how to do it with pari
> >
> > Could anyone give me some hint please?
>
> You are asking for rational point on the curve x^3+y^3 = 22 which is an
> elliptic curve. By posing u=x+y v=x-y, your equation became
> u^3+3*v^2*u = 88
> Then by posing X = 3*88/u; Y = 9*88*v/u you get
> Y^2 = X^3-27*88^2
> which is a Weierstrass equation, which you can use with ellinit:
> E=ellinit([0,0,0,0,-27*88]);

I meant
E=ellinit([0,0,0,0,-27*88^2]);
obviously. The remainder is correct notwithstanding.

> ? elltors(E)
> %11 = [1,[],[]]
> ? ellanalyticrank(E)
> %6 = [1,4.3180855201574550094703927526599064688]
> ? P=ellheegner(E)
> %7 = [553/9,4085/27]
> So you get a point on E. Using ellpow, you can get others:
> ? ellpow(E,P,2)
> %9 = [767848016929/600740100,-672808015029320783/14724139851000]
> ? ellpow(E,P,3)
> %10 = [385268181123102953483527273/4808405283271058302221969,5825874533030960746183337375622326929195/10543907772550436016539105119340117703]

Cheers,
Bill.

```