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NumberTheory

  

ThueSolve

  

solutions to a Thue equation or inequality

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

ThueSolve(expr)

ThueSolve(expr, bound = b)

ThueSolve(expr, vars, bound = b)

Parameters

expr

-

Thue equation or Thue inequality

vars

-

set of two names

bound = b

-

(optional) keyword argument where b is a positive integer; defaults to 10

Description

• 

The ThueSolve function computes all the solutions to a Thue equation or inequality.

• 

Let fx,y be a binary form with integer coefficients and irreducible over the rationals. A binary form is a bivariate polynomial where every term has the same degree. Let m be an integer. A Thue equation has the form fx,y=m and a Thue inequality has the form fx,ym.

• 

If the degree of f is less than 3, then isolve is used to find the solutions. Otherwise, there exists a finite number of solutions and this command finds all solutions x,y given the constraint y10bound.

Examples

with(NumberTheory):

ThueSolve(x^2 + x*y + y^2 = 19);

x=−5,y=2,x=−5,y=3,x=−3,y=−2,x=−3,y=5,x=−2,y=−3,x=−2,y=5,x=2,y=−5,x=2,y=3,x=3,y=−5,x=3,y=2,x=5,y=−3,x=5,y=−2

(1)

The variables may be explicitly given.

ThueSolve(x^3-3*x*y^2+y^3 = 3, {x, y});

x=−1,y=−2,x=−1,y=1,x=2,y=1

(2)

Setting infolevel to 1 or greater will give additional information when solutions do not exist or when solving a Thue inequality.

infolevel[ThueSolve] := 1;

infolevelThueSolve1

(3)

ThueSolve(x^3-3*x*y^2+y^3 = 2);

ThueSolve:   try the following constant terms   -3, -1, 0, 1, 3

(4)

ThueSolve(abs(x^3+x^2*y-2*x*y^2-y^3) <= 5);

ThueSolve:   equality holds for the follow constant terms   0, 1

x=0&comma;y=0&comma;x=−9&comma;y=5&comma;x=−5&comma;y=−4&comma;x=−4&comma;y=9&comma;x=−2&comma;y=1&comma;x=−1&comma;y=−1&comma;x=−1&comma;y=0&comma;x=−1&comma;y=1&comma;x=−1&comma;y=2&comma;x=0&comma;y=−1&comma;x=0&comma;y=1&comma;x=1&comma;y=−2&comma;x=1&comma;y=−1&comma;x=1&comma;y=0&comma;x=1&comma;y=1&comma;x=2&comma;y=−1&comma;x=4&comma;y=−9&comma;x=5&comma;y=4&comma;x=9&comma;y=−5

(5)

Compatibility

• 

The NumberTheory[ThueSolve] command was introduced in Maple 2016.

• 

For more information on Maple 2016 changes, see Updates in Maple 2016.

See Also

isolve

NumberTheory