New Proofs Expand the Limits of What Cannot Be Known

David Hilbert's 10th problem concerns Diophantine equations: polynomials with integer coefficients, such as x2 + y2 = 5 .

In the 1930s , Kurt Gödel demonstrated that this is impossible: In any mathematical system, there are statements that can be neither proved nor disproved.

In 1970 , a Russian mathematician named Yuri Matiyasevich shattered this dream.

Problem of interest is equivalent to famous undecidable problem in computer science called the halting problem.

For every Turing machine, there is a corresponding Diophantine equation for every single ring of integers.

Problem falls apart when equations are allowed to have non-integer solutions, such as y = x2.

Peter Koymans , a mathematician at Utrecht University , has been thinking about Hilbert ’s 10th problem since he was an undergraduate.

The mathematicians needed to get better control over the quadratic twist to solve the problem.

They used a method from a separate area of math called additive combinatorics to ensure that the right combination of primes existed for every ring.