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 x2 − y3 = 7 and x2 + y2 = z2 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.