New Proofs Probe the Limits of Mathematical Truth | Quanta Magazine

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

In 1970 , a Russian mathematician named Yuri Matiyasevich showed that there is no general algorithm that can determine whether any given Diophantastic equation has integer solutions.

Mathematicians suspected that, for every single ring of integers, the problem is still undecidable.

But in 1988 , a graduate student at New York University named Sasha Shlapentokh started to play with ideas for how to get around this problem.

The useful correspondence between Turing machines and Diophantine equations falls apart when the equations are allowed to have non-integer solutions.

Mathematical duo solved Hilbert ’s undecidable 10th problem with elliptic curves.

They used a special equation called an elliptic curve to solve the problem.

But they got stuck trying to get better control over the quadratic twist.

The result was settled Thursday , when an independent team of four mathematicians announced their paper online.