Hilbert's tenth problem

Hilbert's tenth problem is a problem in mathematics that is named after David Hilbert who included it in Hilbert's problems as a very important problem in mathematics. It is about finding an algorithm that can say whether a Diophantine equation has integer solutions. It was proved, in 1970, that such an algorithm does not exist.

Overview

As with all problems included in Hilbert's problems, it was unresolved and considered very difficult when the famous list was published in 1900. The first significant step towards finding the solution was made in 1950 by Julia Robinson, who created a hypothesis (known as the JR hypothesis) around which all later progress was centred. The problem was finally solved with a negative answer in 1970 when Yuri Matiyasevich proved the JR hypothesis. Previous research had shown that this was enough for a negative answer to the problem. The negative answer means that we cannot make an algorithm that can say whether a Diophantine equation has integer solutions.