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

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

On Tue, Aug 02, 2022 at 08:26:03AM +0200, Ralph Beckmann wrote:
> Good morning,
> 
> I am new to PARI and I would like to use it on cubic Pell equations.
> 
> Can anyone show me how to solve x³-ay³=1 (a,x,y ∈ ℕ)?

Hello Ralf,

There are several ways to handle such equations, 
but the simplest with PARI/GP is to consider this as a Thue equation
with P(x,y)=x³-ay³

In PARI/GP this is done as follows
thue(x^3-a,1)

For example for a=37

? thue(x^3-37,1)
%42 = [[1,0],[10,3]]

which meant that the solution are (x,y)=(1,0) and (x,y)=(10,3)
and indeed 10^3-37*3^3=1

Cheers,
Bill.