Parameterized norm form equations with arithmetic progressions

Let be a zero of the Thomas polynomial X 3 (a 1)X 2 (a+2)X 1. We find all algebraic numbers µ = x0+x1 +x2 2 2 Z[ ], such that x0,x1,x2 2 Z forms an arithmetic progression and the norm of µ is less than |2a + 1|. In order to find all progressions we reduce our problem to solve a family of Thue equations and solve this family completely.

[1]  John J. Cannon,et al.  The Magma Algebra System I: The User Language , 1997, J. Symb. Comput..

[2]  E. Matveev,et al.  An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers , 1998 .

[3]  Maurice Mignotte,et al.  Complete solutions of a family of quartic Thue and index form equations , 1996, Math. Comput..

[4]  S. Lang.  Algebraic Number Theory , 1971 .

[5]  L. Vulakh.  Diophantine approximation in ⁿ , 1995 .

[6]  Emery Thomas,et al.  Fundamental units for orders in certain cubic number fields. , 1979 .

[7]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[8]  Emery Thomas,et al.  Complete solutions to a family of cubic Diophantine equations , 1990 .

[9]  S. Lang.  Number Theory III , 1991 .

[10]  A. Pethö,et al.  Simplest cubic fields , 1995 .

[11]  H. Davenport,et al.  THE EQUATIONS 3x2−2 = y2 AND 8x2−7 = z2 , 1969 .

[12]  M. Mignotte,et al.  Verification of a Conjecture of E. Thomas , 1993 .

[13]  A. Pethö,et al.  On norm form equations with solutions forming arithmetic progressions , 2004, Publicationes Mathematicae Debrecen.

[14]  Michel Laurent,et al.  Formes linéaires en deux logarithmes et déterminants d′interpolation , 1995 .

[15]  Clemens Heuberger,et al.  Thomas' Family of Thue Equations Over Imaginary Quadratic Fields , 2002, J. Symb. Comput..

[16]  J. Buchmann,et al.  Computation of independent units in number fields by Dirichlet’s method , 1989 .

[17]  Keith O. Geddes,et al.  Maple 6 Programming Guide , 2000 .