Bill Allombert on Tue, 02 Aug 2022 14:24:00 +0200

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Re: solving x^3-ay^3=1 over integers (N)

On Tue, Aug 02, 2022 at 01:25:11PM +0200, Ralph Beckmann wrote:
> Dear Bill,
> Thank very much for your so prompt and helpful answer.
> Thue works very well!
> Since am just a beginner and math student at the end of my first year, please allow me two more questions:
> (a) When thue() is not finding another solution than the trivial one  - does it mean there is none? 

Yes, if you assume the GRH at least. If you do not want to assume the GRH, you need to to use
thueinit with flag 1.

> Likewise, (b) when Thue finds a non-trivial solution, can I be sure it is the only one?

Yes, idem.

The algorithm used is the Bilu-Hanrot algorithm, see

"Thue Equations of High Degree", by Yu. Bilu and G. Hanrot, J. Number Theory (1996).
"Solving Thue equations without the full unit group", Math. Comp. (2000)

As for the litterature: 

Part of your issue is that nobody calls this a "cubic Pell equation".
Your equation is
- a Thue equation P(x,y)=1 with P=x^3-a*y^3
- a Elliptic curve x^3+z^3 = a*y^3 (with z=-1).
- a norm equation in Z[beta] with beta^3=a, Norm(x+y*beta+z*beta^2)=1 with z=0

(all this you can experiment on with PARI/GP).

I suppose for a first year student, the norm equation approach is the simplest but
it is computionaly flawed.
The Elliptic curve approach is probably the most entertaining, but you get rational

I can explain how to implement those equations in PARI/GP, if you like.