From: rusin@vesuvius.math.niu.edu (Dave Rusin) Newsgroups: sci.math Subject: Re: Equation of degree 3 - rational solution? Date: 29 Jul 1996 17:08:21 GMT In article <4tgjmm$r9l@cri.ens-lyon.fr>, Vincent Lefevre wrote: >Here's the real problem: let p, q, r, s, t be integers and consider >the following equation: pa^3 + qa^2b + rab^2 + sb^3 = t, where a and b >are integers. How can this diophantine equation be solved? > >(note: we can suppose here that the equation px^3 + qx^2 + rx + s = 0 >has no rational solution.) If F(x,y) is an irreducible binary form of degree greater than 2 (yours is of degree 3) then the equation F(x,y) = t is called a Thue equation. It has at most finitely many solutions. There is abundant literature on solving these effectively, but it's not elementary. For example, there is a relevant paper by N Tzanakis and BMM de Weger, J Number Thoery 31 (1989) 99-132. Thue solvers have been incorporated into the major computer algebra programs (e.g. Maple). dave ============================================================================== From: rusin@vesuvius.math.niu.edu (Dave Rusin) Newsgroups: sci.math Subject: Re: Equation of degree 3 - rational solution? Date: 30 Jul 1996 18:53:42 GMT Continuing a thread about Thue equations: In article <4tj434$l4r@cri.ens-lyon.fr>, Vincent Lefevre wrote: >I've just checked Maple. The help says that it only computes the >solutions with the constraint abs(y) < a given constant. But I >need all the solutions. You're right about Maple. Of course, Maple can go through a large solution space quickly for this particular kind of equation -- the default upper bound is 10^10 and can be reached in seconds. There is an effective procedure to add a practical upper bound; I don't know why it's not in Maple. It is available in KASH -- at least, there is nothing in the documentation which suggests the program will only find solutions below some default upper bound. KASH is freely available at this URL: ftp://ftp.math.tu-berlin.de/pub/algebra/Kant/Kash/Binaries There are executables for several workstations. Note that the program comes as a *.tar.gz file, and the documentation is a .dvi file; clearly this program is intended for a researcher's environment. The file is rather large and their server is slow; I don't think there are any mirror sites available (e.g. west of the Atlantic). So you have to really want this program to go after it. Here is the output of the program when supplied with your sample problems (format adjusted slightly). There are no non-trivial solutions: >If someone has a package that can solve the Thue equations, here are >the equations I need to solve: > >2a^3 - 3a^2b - 6ab^2 + b^3 = 2 >2a^3 - 3a^2b - 6ab^2 + b^3 = -2 >a^3 - 9a^2b - 3ab^2 + 3b^3 = 1 >a^3 - 9a^2b - 3ab^2 + 3b^3 = -1 kash> t:=Thue([2, -3, -6, 1]); 2 X^3 - 3 X^2 Y - 6 X Y^2 + Y^3 kash> ThueSolve(t, 2, "abs"); [ [ -1, 0 ], [ -1, 1 ], [ 1, -1 ], [ 1, 0 ] ] kash> u:=Thue([1, -9, -3, 3]); X^3 - 9 X^2 Y - 3 X Y^2 + 3 Y^3 kash> ThueSolve(u,1,"abs"); [ [ -1, 0 ], [ 1, 0 ] ] >There are mail addresses or web sites that can identify sequences >of integers or identify real numbers. Is there a site where some >diophantine equations can be solved? It seems to be the pattern, rather, that once a class of equations is matched with a solution technique, that algorithm is encoded into software which is in turn made available over the net. The newsgroup sci.math.numberthy may be the place to turn with specific requests, or perhaps sci.math.symbolic. dave