The proof of Matiyasevich's theorem solved Hilbert's tenth problem about these entities.
The Mordell conjecture about these entities was solved by Faltings's theorem. The more general Hasse's local-global principle is used to study these objects, one example of which is Bézout's identity. A method named for Størmer may be applied to their exponential types, such as that which appears in Catalan's conjecture. The proof of Matiyasevich's theorem solved Hilbert's tenth problem about these entities. Pell's equation is an example of, for 10 points, what polynomial equations whose variables take integer values, named after a third century mathematician from Greece?
The Mordell conjecture about these entities was solved by Faltings's theorem.
{Diophantine} equations