Diophantine problems

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A Diophantine problem asks for solutions over a discrete set to a system of equations

$f_{i}(x_{1},\ldots ,x_{n})=0,\quad i=1,\ldots ,m,\,$

where the discrete set (for Diophantus, it was the positive rational numbers) could be

• a ring of integers (e.g. the rational integers $\mathbb {Z}$),
• a number field (e.g. the rational numbers $\mathbb {Q}$).

See also

• Hilbert’s 10th problem (Matiyasevich's theorem)

External links

• H. P. F. Swinnerton-Dyer, Diophantine Problems, Distinguished Lecture Series (Lecture 1).