Re: S-Unit questions

[hermann@stamm-wilbrandt.de]

On 2026-03-12 15:10, Bill Allombert wrote:
> On Thu, Mar 12, 2026 at 02:58:16PM +0100, hermann@stamm-wilbrandt.de 
> wrote:
> > On error message it explained that linear polynomial has too low 
> > degree for
> > thueinit.
> > And to use a script with "bnfisunit()" — but current GP does not know 
> > this
> > command.
>
> bnfisunit certainly exist, though it will probably not help you...
>
> ? ?bnfisunit()
> bnfisunit(bnf,x,{U}): bnf being output by bnfinit, give the column
> vector of exponents of x on the fundamental units and the roots of
> unity if x is a unit, the empty vector otherwise. If U is present, as
> given by bnfunits, decompose x on the attached S-units generators.
Sorry, I coorected that in my 2nd email.

> > So how to solve this equation using PARI/GP with its Baker(?) 
> > algorithm?
> >
> > ±2^{a1}*3^{b1}±2^{a2}*3^{b2}=1
>
> I do not think there is a GP function that can solve this, though this
> is solvable
> in principle using linear forms in logarithms.
>
> Cheers,
> Bill.
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>


Regards,

Hermann.