The List of Hilbert’s Twenty-Three Problems: Hilbert’s Tenth Problem

Hilbert’s tenth problem concerns finding an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution.

Polynomial equations in a finite number of variables with integer coefficients are known as Diophantine equations. Equations like 

$x^2 − y^3 = 7$ and $x^2 + y^2 = z^2$ 

are examples. For centuries, mathematicians have wondered whether certain Diophantine equations have integer solutions. Hilbert’s 10th problem asks whether there is an algorithm to determine whether a given Diophantine equation has integer solutions or not. 

In 1970, Yuri Matiyasevich completed a proof that no such algorithm exists.

Πηγή: abakcus

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