Re: S-Unit questions

[Aurel Page <aurel.page@normalesup.org>]

On 3/12/26 15:36, hermann@stamm-wilbrandt.de wrote:
> > > So how to solve this equation using PARI/GP with its Baker(?) 
> > > algorithm?
> > >
> > > ±2^{a1}*3^{b1}±2^{a2}*3^{b2}=1
> This is the script proposed by Gemini, which errors on bnfuniteqn() 
> not existing:
>
> ? {
>   \\ 1. Initialize Q
>   bnf = bnfinit(x - 1);
>
>   \\ 2. Define S = {2, 3} correctly
>   S = concat([idealprimedec(bnf, 2), idealprimedec(bnf, 3)]);
>
>   \\ 3. Get the S-unit structure AND the map (the '1' at the end is 
> vital)
>   s_struct = bnfsunit(bnf, S);
>
>   \\ 4. Solve the S-unit equation epsilon_1 + epsilon_2 = 1
>   \\ This returns the solutions as exponent vectors relative to the 
> S-unit generators
>   solutions_vecs = bnfuniteqn(s_struct);
>
>   \\ 5. Convert exponent vectors back into actual rational numbers
>   \\ s_struct[1] contains the generators: [-1, 2, 3]
>   gens = s_struct[1];
>
>   print("Solutions (x, y) where x + y = 1:");
>   for(i = 1, #solutions_vecs,
>       sol = solutions_vecs[i];
>       \\ The solution is a pair [x, y]
>       x = nfeltfactorback(bnf, gens, sol[1]);
>       y = nfeltfactorback(bnf, gens, sol[2]);
>       print([x, y]);
>   );
> }
>   ***   at top-level: ...it(bnf,S);solutions_vecs=bnfuniteqn(s_struct);
>   *** ^---------------------
>   ***   not a function in function call
>   ***   Break loop: type 'break' to go back to GP prompt
> break>
This solution is a hallucination: we do not currently have an S-unit 
equation solver in PARI, though it would be interesting to have one.

Cheers,
Aurel