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

[Bill Allombert <Bill.Allombert@math.u-bordeaux1.fr>]

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]);
? 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]
etc.

Now of course, you have to recover the solution of the original equation from X and Y.
You get for example:
14176649488890906071282308630139700149971^3+2524900422828984578915633878895046291581^3
=22*5068914033007802762941623871038239169306^3

Cheers,
Bill.