Thue equation
Encyclopedia
In mathematics
, a Thue equation is a Diophantine equation
of the form
where ƒ is an irreducible
bivariate form of degree at least 3 over the rational numbers, and r is a nonzero rational number
. It is named after Axel Thue
who in 1909 proved a theorem
, now called Thue's theorem, that Thue equation has finitely many solutions in integers x and y.
s:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Thue equation is a Diophantine equation
Diophantine equation
In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...
of the form
- ƒ(x,y) = r,
where ƒ is an irreducible
Irreducible polynomial
In mathematics, the adjective irreducible means that an object cannot be expressed as the product of two or more non-trivial factors in a given set. See also factorization....
bivariate form of degree at least 3 over the rational numbers, and r is a nonzero rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
. It is named after Axel Thue
Axel Thue
Axel Thue was a Norwegian mathematician, known for highly original work in diophantine approximation, and combinatorics....
who in 1909 proved a theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
, now called Thue's theorem, that Thue equation has finitely many solutions in integers x and y.
Solving Thue equations
Solving a Thue equation can be described as an algorithm ready for implementation in software. In particular, it is implemented in the following computer algebra systemComputer algebra system
A computer algebra system is a software program that facilitates symbolic mathematics. The core functionality of a CAS is manipulation of mathematical expressions in symbolic form.-Symbolic manipulations:...
s:
- in PARI/GPPARI/GPPARI/GP is a computer algebra system with the main aim of facilitating number theory computations. It is free software; versions 2.1.0 and higher are distributed under the GNU General Public License...
as functions thueinit and thue.