Diophantine equation

In algebraic number theory, a Diophantine equation is ( named after the Greek mathematician Diophantus of Alexandria, 250 ) an equation of the form ( polynomial with integer coefficients), in which one is only interested in integer solutions. This limitation of the amount of solution makes sense if Teilbarkeitsfragen to be answered when it comes to problems of Kongruenzarithmetik or when problems in practice only integer solutions are meaningful, for example, the number distribution in the manufacture of several products.

Examples

  • Has as a solution the number pairs (1,1), ( -1,1 ), ( 2,4 ), ( -2,4 ), ( 3,9 ), ( -3,9 ), ... in general: ( ± n, n2 ).
  • Has no solution, since the left side of the equation is always greater than zero.
  • Has no solution, since Diophantine equations only integer solutions are sought.

Linear Diophantine Equation

Diophantine equations in which appear not powers, they are called linear. For them, there are algorithms that ( after finitely many steps ) will always find all solutions.

Famous Diophantine equations

Pythagorean triples

The integer solutions of the so-called Pythagorean triples form. They are found mainly due to the approach.

Fermat's Last Theorem

If one were to generalize the above equation, one obtains a diophantine equation. As Fermat's last theorem is called the set up by Pierre de Fermat 400 years ago, claiming that it has no integer solution for ( non-trivial solutions in which one of the numbers is zero ), which was proved in 1994 by Andrew Wiles.

Pellsche equation

In addition to the linear equations, the so-called Diophantine equation Pellsche

Particularly important being the smallest value pair is to search for a given, from which all other pairs can easily find them. When a square number, this equation, with the exception of the trivial solution, and not an integer detachably. The resolution of the Pell equation is equivalent to the prospect of the units in the ring of algebraic integers of the body that arises from the rational number field by adjoining the square root of.

Hilbert's Tenth Problem

In 1900, David Hilbert presented the problem of solvability of a diophantine equation as the tenth issue of his famous list of 23 mathematical problems. 1970 proved Yuri Vladimirovich Matijassewitsch that the solvability of a Diophantine equation is undecidable.

241251
de