{"id":14331,"date":"2025-03-10T01:05:47","date_gmt":"2025-03-10T01:05:47","guid":{"rendered":"https:\/\/topat10.com\/?p=14331"},"modified":"2025-03-10T01:05:47","modified_gmt":"2025-03-10T01:05:47","slug":"new-proofs-expand-the-limits-of-what-cannot-be-known","status":"publish","type":"post","link":"https:\/\/topat10.com\/?p=14331","title":{"rendered":"New Proofs Expand the Limits of What Cannot Be Known"},"content":{"rendered":"


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In other words, Hilbert\u2019s 10th problem is undecidable.<\/p>\n

Mathematicians hoped to follow the same approach to prove the extended, rings-of-integers version of the problem\u2014but they hit a snag.<\/p>\n

Gumming Up the Works<\/h2>\n

The useful correspondence between Turing machines and Diophantine equations falls apart when the equations are allowed to have non-integer solutions. For instance, consider again the equation y<\/em> = x<\/em>2<\/sup>. If you\u2019re working in a ring of integers that includes \u221a2, then you\u2019ll end up with some new solutions, such as x<\/em> = \u221a2, y<\/em> = 2. The equation no longer corresponds to a Turing machine that computes perfect squares\u2014and, more generally, the Diophantine equations can no longer encode the halting problem.<\/p>\n

But in 1988, a graduate student at New York University named Sasha Shlapentokh<\/a> started to play with ideas for how to get around this problem. By 2000, she and others had formulated a plan. Say you were to add a bunch of extra terms to an equation like y<\/em> = x<\/em>2<\/sup> that magically forced x<\/em> to be an integer again, even in a different number system. Then you could salvage the correspondence to a Turing machine. Could the same be done for all Diophantine equations? If so, it would mean that Hilbert\u2019s problem could encode the halting problem in the new number system.<\/p>\n

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