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  "body": "\n## Definition\n\nLet $d$ be a positive integer that is not a perfect square.\nThe fundamental solution of [[Pell's equation|0AF3DANJ76]] $x^2 - d y^2 = 1$ is the solution $(x_1, y_1)$ with $x_1 \u003e 1$ and $y_1 \u003e 0$ such that $x_1$ is as small as possible.\n\n## Remarks\n\nThe fundamental solution exists by the [[existence of nontrivial solutions|0EM6C9SZN5]].\nAmong the solutions with $x \u003e 1$ and $y \u003e 0$, it also has the smallest $y$ and the smallest value of $x + y \\sqrt{d}$.\nFor example, the fundamental solution for $d = 2$ is $(3, 2)$, and for $d = 7$ it is $(8, 3)$.\nIt appears among the convergents of the continued fraction of $\\sqrt{d}$.\n\n## Explanations\n\n### L1 human\n\nThe fundamental solution is the smallest solution with positive entries.\nAll other solutions come from it by taking [[powers|0J1TT091TN]] of $x_1 + y_1 \\sqrt{d}$.\n",
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