Re: S-Unit questions

On 2026-03-12 15:10, Bill Allombert wrote:
> On Thu, Mar 12, 2026 at 02:58:16PM +0100, [email protected] 
> 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.