DIOPHANTINE EQUATION

Speaking about Diophantine equation, one usually has in mind two things:

• particular very simple form of such an equation, namely,
  

  P(a1,...,an,x1,...,xm) = 0,

  (1)

  where P is a polynomial;

• the admissible values of the unknowns; usually they can range over integers or over natural numbers, but other domains, like rational or Gaussian numbers, can be also considered.

In number-theoretical literature Diophantine equations are sometime treated in a broader sense by taking into account only the domain of admissible values of the unknowns. In such a case no distinction is made, for example, between the genuine Diophantine equations and so-called exponential Diophantine equations.

One has also to specify what could be the coefficients of the polynomial in (1). Usually they are allowed to be arbitrary integers, but when the domain of unknowns is broader than the set of integers, it is sometimes natural to allow the coefficients to belong to the set of admissible values of the unknowns. (See also Diophantine sets over other domains.)

A system of two or more Diophantine equations can be usually combined into one. This is evident when the coefficients and the unknowns range over integers, namely, system

P1(x1,...,xm) = 0 ,

P2(x1,...,xm) = 0

is clearly equivalent to

P12(x1,...,xm) + P22(x1,...,xm) = 0 .

(2)

More care should be taken when coefficients or unknowns can have other values.

Besides single Diophantine equations we can consider families of Diophantine equations or parametric Diophantine equations. They have the form of a polynomial equation

P(a1,...,an,x1,...,xm) = 0 ,

(3)

in which the variables are divided into

• parameters a₁,...,a_n;
• unknowns x₁,...,x_m.

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