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  "body": "\n## Statement\n\nLet $d$ be a positive integer that is not a perfect square, and let $(x_1, y_1)$ be the [[fundamental solution|0KCYJSB9FW]] of $x^2 - d y^2 = 1$.\nLet $x$ and $y$ be integers with $x \\ge 1$, $y \\ge 0$ and $x^2 - d y^2 = 1$.\nThen there is a unique natural number $n$ with $x + y \\sqrt{d} = (x_1 + y_1 \\sqrt{d})^n$.\n\n## Remarks\n\nConversely, each power $(x_1 + y_1 \\sqrt{d})^n = x_n + y_n \\sqrt{d}$ gives a solution $(x_n, y_n)$.\nThe integer solutions are exactly the pairs with $x + y \\sqrt{d} = \\pm (x_1 + y_1 \\sqrt{d})^n$ for $n \\in \\mathbb{Z}$.\nFor $d = 2$: $(3 + 2\\sqrt{2})^2 = 17 + 12 \\sqrt{2}$, so $(17, 12)$ is the next solution after $(3, 2)$.\nIn algebraic terms, the units of norm $1$ in $\\mathbb{Z}[\\sqrt{d}]$ form the group $\\{\\pm 1\\} \\times \\mathbb{Z}$.\n\n## Explanations\n\n### L1 human\n\nOnce you know the smallest solution, you get all the others by taking powers.\nExpand $(x_1 + y_1 \\sqrt{d})^n$ and read the integer part and the coefficient of $\\sqrt{d}$.\n",
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