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  "body": "\n## Definition\n\nLet $n \\ge 3$ be an integer and let\n$$f(x, y) = a_n x^n + a_{n-1} x^{n-1} y + \\dots + a_0 y^n$$\nbe a homogeneous polynomial of degree $n$ with integer coefficients $a_0, \\dots, a_n$.\nAssume that $f$ is [[irreducible|03FKX7Z5DC]] over $\\Q$.\nLet $m$ be a nonzero integer.\nThe *Thue equation* for $f$ and $m$ is the [[Diophantine equation|08B1RTW7VZ]] $f(x, y) = m$ in the unknowns $x, y \\in \\Z$.\n\n## Remarks\n\nThue's theorem says that a Thue equation has only finitely many solutions $(x, y) \\in \\Z^2$.\n\nExample: $x^3 - 2y^3 = 1$ is a Thue equation, since $x^3 - 2y^3$ is homogeneous of degree $3$ and irreducible over $\\Q$, because $2$ is not the cube of a rational number.\nIt has the solutions $(1, 0)$ and $(-1, -1)$: indeed $(-1)^3 - 2(-1)^3 = 1$.\n\nNon-example: $x^2 - 2y^2 = 1$ is not a Thue equation, because the degree is $2$.\nIt is a Pell equation, and it has infinitely many solutions, for example $(3, 2)$ and $(17, 12)$.\n",
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