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  "body": "\n## Summary\n\nA Diophantine equation is a polynomial equation with integer coefficients, for which one looks for integer solutions, or sometimes rational solutions.\nTypical questions ask whether a solution exists, whether there are finitely many, and how to find all of them.\n\n## Remarks\n\nExamples are the linear equation $ax + by = c$, [[Pythagorean triples|01G1WY1GVX]] $x^2 + y^2 = z^2$, [[Pell's equation|0AF3DANJ76]] $x^2 - d y^2 = 1$ and the equation of [[Fermat's Last Theorem|0Y5X3SDBCZ]].\nBy Hilbert's tenth problem, as solved by Yuri Matiyasevich in 1970, no algorithm decides for every Diophantine equation whether it has an integer solution.\nThe name refers to Diophantus of Alexandria.\n\n## Explanations\n\n### L1 human\n\nIn a Diophantine equation, only whole numbers count as solutions.\nFor example, $x^2 + y^2 = 3$ has real solutions, but no integer solutions.\nThe restriction to integers makes such equations much harder than ordinary equations.\n",
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