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New Proofs Expand the Limits of What Cannot Be Known
By proving a broader version of Hilbert’s famous 10th problem, two groups of mathematicians have expanded the realm of mathematical unknowability.
Both groups have proved that, for a vast and important collection of settings beyond integers, there is likewise no general algorithm to determine if any given Diophantine equation has a solution. Over the years, Shlapentokh and other mathematicians figured out what terms they had to add to the Diophantine equations for various kinds of rings, which allowed them to demonstrate that Hilbert’s problem was still undecidable in those settings. This allowed them to apply a method from an entirely separate area of math, called additive combinatorics, to ensure that the right combination of primes existed for every ring.
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