Hilbert's Tenth Problem

An equation such as $x^{2} + y^{2} = 25$ can have whole-number solutions, such as $x = 3$ and $y = 4$ . Both of its sides are polynomials. A polynomial is a formula that only adds and multiplies numbers and variables. In 1900, David Hilbert published a famous list of 23 open problems. The tenth asks for a method that takes any polynomial equation built from whole numbers and decides if it has a whole-number solution. A method here means fixed steps that a computer could follow.

Some cases are easy. The equation $x^{2} + y^{2} = 3$ has no whole-number solution. Squares are 0, 1, 4, 9 and so on, and no two of them add up to 3. Other cases are very hard. The equation $x^{3} + y^{3} + z^{3} = 42$ has a solution with 17-digit numbers, and a large computer search found it only in 2019.

A search is not enough. A computer can try numbers one after another, and it will find a solution if one exists. But if no solution exists, the search runs forever, and you never learn that the answer is no.

In 1970, mathematicians proved that no method exists for whole numbers. The reason is surprising. Any computer program can be turned into an equation that has a whole-number solution exactly when the program eventually stops. Alan Turing had proved that no program can always tell if another program stops. So no program can settle all equations.

This does not mean that a single equation has no answer. Each one has a definite answer, and special tricks solve many of them. It means that no single method gets all of them right.

Equation	Whole numbers	Fractions
$2 x = 1$	None	$1 / 2$
$x^{2} + y^{2} = 1$	$(1, 0)$	$(3 / 5, 4 / 5)$
$x^{2} + y^{2} = 3$	None	None
$x^{3} + y^{3} = 6$	None	$(17 / 21, 37 / 21)$

The new result asks the same question for fractions. Fractions change the answers, as the table shows. The equation $2 x = 1$ has no whole-number solution, but it has the fraction $x = 1 / 2$ .

Fraction solutions can also be enormous. For $y^{2} = x^{3} + 877 x$ , take the simplest solution with $x$ not zero. The top number of the fraction $x$ has 42 digits. So a search can miss an answer for a very long time.

The 1970 argument does not transfer directly, because its equations need whole-number solutions. A fraction that solves the equation says nothing about the program. So the question for fractions stayed open for decades.

An OpenAI paper reports that the answer is still no. No method can decide, for each polynomial equation built from whole numbers, if it has a fraction solution. This holds even when the highest power in the equation is at most 4. A later paper reports a more direct route.