By proving a broader version of Hilbert’s famous 10th problem, two groups of mathematicians have expanded the realm of mathematical unknowability.
The post New Proofs Probe the Limits of Mathematical Truth first appeared on Quanta Magazine
Summary
- In 1900, the eminent mathematician David Hilbert announced 23 key problems to resolve within mathematics, one of which was whether it was possible to determine whether any given Diophantine equation (a polynomial with integer coefficients) has integer solutions, a concept known as Hilbert’s 10th problem.
- In 1970, mathematician Yuri Matiyasevich resolved the problem, showing there was no general algorithm to determine whether any given Diophantine equation has integer solutions, meaning the problem is unsolvable.
- Mathematicians have subsequently attempted to extend this problem to other number systems and prove that it is likewise unsolvable in those settings, and now, two teams of mathematicians have independently proven this to be the case for number systems related to rings of integers, extending the scope of the problem even further.