ℤ is Diophantine in ℚ

This title uses two symbols. $ℤ$ means the whole numbers, such as $- 2$ , $0$ , $1$ and $3$ . $ℚ$ means the fractions, such as $1 / 2$ and $- 3 / 4$ . Every whole number is also a fraction, because $3$ is the same as $3 / 1$ .

A polynomial is a formula that only adds and multiplies numbers and variables. For example, $x^{2} + 3 x y - 5$ is a polynomial. A polynomial equation sets such a formula equal to zero.

You can use an equation as a test. You have a number $q$ , and the equation has $q$ and some extra variables in it. The number $q$ passes if some fractions for the extra variables make the equation true. If no fractions work, $q$ fails.

Here is an example, with the equation $q w - 1 = 0$ . The number $3 / 4$ passes, because $w = 4 / 3$ gives $(3 / 4) (4 / 3) - 1 = 0$ . The number $0$ fails, because $0$ times any $w$ is $0$ , and $0 - 1$ is never $0$ . So this equation tests if a number is not zero.

Here is a second example, with the equation $q - w^{2} = 0$ . The number $9 / 4$ passes, because $w = 3 / 2$ works. The number $2$ fails, because no fraction squared gives $2$ . So this equation tests if a number is the square of a fraction.

A set of fractions is Diophantine in $ℚ$ if one equation like this can test for it. The name comes from Diophantus, a Greek mathematician who studied equations. So the nonzero numbers are Diophantine in $ℚ$ , and so are the squares.

The title says that the whole numbers are also Diophantine in $ℚ$ . One equation exists that every whole number passes and every other fraction fails. So $3$ and $- 12$ pass, and $7 / 2$ and $1 / 3$ fail.

The curve

q = w^{2}

. Each dot has a fraction for

w

. Go straight down from a dot, and you land on the square of a fraction, such as

1 / 4

1

9 / 4

. Nothing lands on

2

This is surprising. You can see that $7 / 2$ is not a whole number by looking at its bottom number. An equation cannot do that. It can only add and multiply.

There was also a strong reason to doubt it. The solutions of an equation form a curve or a surface, and the figure shows one. Fraction solutions usually crowd together along such a shape, so the points directly below them on the line crowd together too. But the whole numbers sit apart, with a gap of 1 between neighbors. A conjecture of Barry Mazur would have ruled out such an equation.

A new paper reports that the equation exists, and it publishes all of it. The equation has 12,504 variables: $q$ and 12,503 extra variables. Its highest power is 4, it has 135,721 terms, and its largest number has 24 digits.

The equation is a sum of 11,939 squares, set equal to zero. The square of a fraction is never negative, so the sum is zero only if every square is zero. So one equation holds 11,939 separate conditions, and the highest power in each condition is 2.

Here is what it gives. A question about whole numbers becomes a question about fractions. To ask for a whole number $x$ , ask for a fraction $x$ that also passes the test. This connects to Hilbert's tenth problem, which asks if a computer can decide such questions.