New Proofs Expand the Limits of What Cannot Be Known

Other Diophantine equations, such as x2 + y2 = 3, don’t have any integer solutions. Hilbert’s 10th problem asked whether it’s always possible to tell if a given Diophantine equation has integer solutions. Does an algorithm exist to determine this for every equation, or is the problem undecidable? There might be no hope for a complete and systematic approach to all of mathematics—or even all 23 of Hilbert’s problems—but one might still exist when it comes to Diophantine equations, forming a microcosm of his original program. “This problem is a very natural version of that dream,” said Peter Koymans of Utrecht University. In 1970, a Russian mathematician named Yuri Matiyasevich shattered this dream. He showed that there is no general algorithm that can determine whether any given Diophantine equation has integer solutions—that Hilbert’s 10th is an undecidable problem. You might be able to come up with an algorithm that can assess most equations, but it won’t work for every single one. ... Read full article.