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numtheory(deprecated)

  

thue

  

solve a Thue equation or inequality

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

thue(ex, vars, k, w)

thue(ex, vars, k)

thue(ex, vars, w)

thue(ex, vars)

Parameters

ex

-

Thue equation or Thue inequality

vars

-

list of two names, [x, y]

k

-

(optional) positive integer

w

-

(optional) unevaluated variable

Description

• 

Important: The numtheory package has been deprecated.  Use the superseding command NumberTheory[ThueSolve] instead.

• 

Let fx&comma;y be a rational binary form in Zx,y, irreducible over Qx,y, and m an integer. The Thue equation has the form fx&comma;y=m and the Thue inequality has the form fx&comma;ym  (with 0<m).

• 

This function computes all solutions in × of the given Thue equation or inequality ex, subject to the constraint y10k.  If k is omitted, it defaults to 10.

• 

In the equation case, if there are no solutions with the above constraint and the last argument to thue is an unevaluated name (i.e., w is given), then to it will be assigned an expression sequence consisting of a diagnostic message and the integers m' in the range 2m..2m such that the given Thue equation has a solution when m is replaced by m'.

• 

In the inequality case, if the last argument to thue is an unevaluated name (i.e., w is given), then w is assigned the set of values obtained by substituting the various solutions to the inequality into the left hand side of the given Thue inequality.

• 

Note that in the inequality case, ex must be given in the form fm and not simply fm.

• 

This function is part of the numtheory package, and so can be used in the form thue(..) only after performing the command with(numtheory). The function can always be accessed in the long form numtheory[thue](..).

Examples

Important: The numtheory package has been deprecated.  Use the superseding command NumberTheory[ThueSolve] instead.

withnumtheory&colon;

thuex5y=1&comma;x&comma;y

x=1+5_Z1&comma;y=_Z1

(1)

thuex2+xy+y2=19&comma;x&comma;y

x=−5&comma;y=2,x=−5&comma;y=3,x=−3&comma;y=−2,x=−3&comma;y=5,x=−2&comma;y=−3,x=−2&comma;y=5,x=2&comma;y=−5,x=2&comma;y=3,x=3&comma;y=−5,x=3&comma;y=2,x=5&comma;y=−3,x=5&comma;y=−2

(2)

fx33xy2+y3

fx33xy2+y3

(3)

thuef=3&comma;x&comma;y

x=−1&comma;y=−2,x=−1&comma;y=1,x=2&comma;y=1

(4)

thuef=2&comma;x&comma;y&comma;w

w

try the following constant term(s): ,−3,−1,0,1,3

(5)

thuef=3&comma;x&comma;y&comma;20&comma;w

x=−2&comma;y=−1,x=1&comma;y=−1,x=1&comma;y=2

(6)

w

−3

(7)

fabsx3+x2y2xy2y3&colon;

solsthuef5&comma;x&comma;y&comma;w

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

(8)

w

equality holds for the following constant term(s): ,0,1

(9)

evalsubssols2&comma;f

1

(10)

gx5+x4y4x3y23x2y3+3xy4+y5

gx5+x4y4x3y23x2y3+3xy4+y5

(11)

thueabsg10&comma;x&comma;y&comma;w

x=0&comma;y=0,x=−2&comma;y=1,x=−1&comma;y=−1,x=−1&comma;y=0,x=−1&comma;y=1,x=0&comma;y=−1,x=0&comma;y=1,x=1&comma;y=−1,x=1&comma;y=0,x=1&comma;y=1,x=2&comma;y=−1

(12)

w

equality holds for the following constant term(s): ,0,1

(13)

Note that the binary forms must be irreducible:

thueabsx3+y35&comma;x&comma;y

Error, (in numtheory:-thue) this binary form is not irreducible

See Also

isolve

NumberTheory[ThueSolve]

numtheory(deprecated)