Re: S-Unit questions
Charles Greathouse <[email protected]> Sat, 14 Mar 2026 10:23:52 -0400
| Newsgroups | gmane.comp.mathematics.pari.user |
|---|---|
| Message-ID | <CANXmBjwGK0Wr8+V5evfvpuhuUEkMv-dk++5U7zQ=ss+2zvEoZQ@mail.gmail.com> |
--000000000000cfdeb5064cfcbc15 Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable I think an efficient S-unit solver would be very useful (I=E2=80=99d use it= , at least), but it seems like a huge ask. On Sat, Mar 14, 2026 at 5:48=E2=80=AFAM <[email protected]> wrote: > Below SageMath solver run for S=3D{2,3} took less than 5 minutes. > I tried with S=3D{2,3,5,7} and aborted after 22.5h on AMD 7950X CPU with > boost frequency 5.4GHz run. > > https://gist.github.com/Hermann-SW/b538c991c922b1e8fc06abbab554b3ef?perma= link_comment_id=3D6027812#gistcomment-6027812 > Then I tried with S=3D{2,3,5} and aborted after 13.5h. > > Since SageMath S-Unit equation solver is not efficient, > an efficient PARI/GP solution somehow would be desirable: > https://en.wikipedia.org/wiki/S-unit#S-unit_equation > > Regards, > > Hermann. > > On 2026-03-12 21:56, [email protected] wrote: > > On 2026-03-12 16:09, [email protected] wrote: > >> > >> The number of solutions to S-Unit equation (u+v=3D1, u,v S-Units) > >> https://en.wikipedia.org/wiki/S-unit#S-unit_equation > >> > >> is finite. That section also says such a solver can be found here, I > >> will try: > >> https://en.wikipedia.org/wiki/SageMath > >> > > Never used SageMath before, and it took many iterations with Gemini to > > get error free code. > > I changed to use =E2=84=9A instead of =E2=84=9A(i) and added generator = output: > > https://gist.github.com/Hermann-SW/b538c991c922b1e8fc06abbab554b3ef > > > > hermann@7950x:~$ time Downloads/SageMath-10.8-x86_64.AppImage > > S-Unit.sol.sage > > Generators: > > -1 > > 2 > > 3 > > Found 11 solutions: > > 2 + -1 =3D=3D 1 > > -------------------- > > -1/8 + 9/8 =3D=3D 1 > > -------------------- > > 1/9 + 8/9 =3D=3D 1 > > -------------------- > > 3 + -2 =3D=3D 1 > > -------------------- > > 4 + -3 =3D=3D 1 > > -------------------- > > 3/4 + 1/4 =3D=3D 1 > > -------------------- > > 1/3 + 2/3 =3D=3D 1 > > -------------------- > > 4/3 + -1/3 =3D=3D 1 > > -------------------- > > -1/2 + 3/2 =3D=3D 1 > > -------------------- > > -8 + 9 =3D=3D 1 > > -------------------- > > 1/2 + 1/2 =3D=3D 1 > > -------------------- > > > > real 4m45.409s > > user 4m46.018s > > sys 0m0.187s > > hermann@7950x:~$ > > > > > > Regards, > > > > Hermann. > > --000000000000cfdeb5064cfcbc15 Content-Type: text/html; charset="UTF-8" Content-Transfer-Encoding: quoted-printable <div dir=3D"auto">I think an efficient S-unit solver would be very useful (= I=E2=80=99d use it, at least), but it seems like a huge ask.</div><div><br>= <div class=3D"gmail_quote gmail_quote_container"><div dir=3D"ltr" class=3D"= gmail_attr">On Sat, Mar 14, 2026 at 5:48=E2=80=AFAM <<a href=3D"mailto:h= [email protected]">[email protected]</a>> wrote:<br></d= iv><blockquote class=3D"gmail_quote" style=3D"margin:0px 0px 0px 0.8ex;bord= er-left-width:1px;border-left-style:solid;padding-left:1ex;border-left-colo= r:rgb(204,204,204)">Below SageMath solver run for S=3D{2,3} took less than = 5 minutes.<br> I tried with S=3D{2,3,5,7} and aborted after 22.5h on AMD 7950X CPU with <b= r> boost frequency 5.4GHz run.<br> <a href=3D"https://gist.github.com/Hermann-SW/b538c991c922b1e8fc06abbab554b= 3ef?permalink_comment_id=3D6027812#gistcomment-6027812" rel=3D"noreferrer" = target=3D"_blank">https://gist.github.com/Hermann-SW/b538c991c922b1e8fc06ab= bab554b3ef?permalink_comment_id=3D6027812#gistcomment-6027812</a><br> Then I tried with S=3D{2,3,5} and aborted after 13.5h.<br> <br> Since SageMath S-Unit equation solver is not efficient,<br> an efficient PARI/GP solution somehow would be desirable:<br> <a href=3D"https://en.wikipedia.org/wiki/S-unit#S-unit_equation" rel=3D"nor= eferrer" target=3D"_blank">https://en.wikipedia.org/wiki/S-unit#S-unit_equa= tion</a><br> <br> Regards,<br> <br> Hermann.<br> <br> On 2026-03-12 21:56, <a href=3D"mailto:[email protected]" target= =3D"_blank">[email protected]</a> wrote:<br> > On 2026-03-12 16:09, <a href=3D"mailto:[email protected]" tar= get=3D"_blank">[email protected]</a> wrote:<br> >> <br> >> The number of solutions to S-Unit equation (u+v=3D1, u,v S-Units)<= br> >> <a href=3D"https://en.wikipedia.org/wiki/S-unit#S-unit_equation" r= el=3D"noreferrer" target=3D"_blank">https://en.wikipedia.org/wiki/S-unit#S-= unit_equation</a><br> >> <br> >> is finite. That section also says such a solver can be found here,= I <br> >> will try:<br> >> <a href=3D"https://en.wikipedia.org/wiki/SageMath" rel=3D"noreferr= er" target=3D"_blank">https://en.wikipedia.org/wiki/SageMath</a><br> >> <br> > Never used SageMath before, and it took many iterations with Gemini to= <br> > get error free code.<br> > I changed to use =E2=84=9A instead of =E2=84=9A(i) and added generator= output:<br> > <a href=3D"https://gist.github.com/Hermann-SW/b538c991c922b1e8fc06abba= b554b3ef" rel=3D"noreferrer" target=3D"_blank">https://gist.github.com/Herm= ann-SW/b538c991c922b1e8fc06abbab554b3ef</a><br> > <br> > hermann@7950x:~$ time Downloads/SageMath-10.8-x86_64.AppImage <br> > S-Unit.sol.sage<br> > Generators:<br> > -1<br> > 2<br> > 3<br> > Found 11 solutions:<br> > 2 + -1 =3D=3D 1<br> > --------------------<br> > -1/8 + 9/8 =3D=3D 1<br> > --------------------<br> > 1/9 + 8/9 =3D=3D 1<br> > --------------------<br> > 3 + -2 =3D=3D 1<br> > --------------------<br> > 4 + -3 =3D=3D 1<br> > --------------------<br> > 3/4 + 1/4 =3D=3D 1<br> > --------------------<br> > 1/3 + 2/3 =3D=3D 1<br> > --------------------<br> > 4/3 + -1/3 =3D=3D 1<br> > --------------------<br> > -1/2 + 3/2 =3D=3D 1<br> > --------------------<br> > -8 + 9 =3D=3D 1<br> > --------------------<br> > 1/2 + 1/2 =3D=3D 1<br> > --------------------<br> > <br> > real=C2=A0 4m45.409s<br> > user=C2=A0 4m46.018s<br> > sys=C2=A0 =C2=A00m0.187s<br> > hermann@7950x:~$<br> > <br> > <br> > Regards,<br> > <br> > Hermann.<br> <br> </blockquote></div></div> --000000000000cfdeb5064cfcbc15--