Abstract
Diophantine equations have been studied throughout the history of mathematics. Initially considered by Diophantus of Alexandria, a Hellenistic mathematician from the third century CE, they include the famous Fermat’s Last Theorem, stated by Pierre de Fermat in the seventeenth century and eventually proved by Andrew Wiles in 1994. Hilbert’s tenth problem asked to provide an algorithm or procedure to decide whether a given Diophantine equation admits integer solutions. Such an algorithm was shown not to exist by Yuri Matiyasevich in 1970, building on previous work of Martin Davis, Hilary Putnam, and Julia Robinson.
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Notes
- 1.
The general form of Rado’s Theorem for a single equation states that c 1 X 1 + ⋯ + c k X k = 0 is partition regular if and only there exists a nonempty set of indexes I ⊆{1, …, k} with ∑i ∈ I c i = 0.
- 2.
This theorem is essentially the content of Exercise 4.11.
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Nasso, M.D., Goldbring, I., Lupini, M. (2019). Partition Regularity of Equations. In: Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory. Lecture Notes in Mathematics, vol 2239. Springer, Cham. https://doi.org/10.1007/978-3-030-17956-4_9
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