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  "body": "\n## Definition\n\nAn infinite [[simple continued fraction|06XF7BCFFJ]] $[a_0; a_1, a_2, \\dots]$ is periodic if there are integers $N \\ge 0$ and $k \\ge 1$ with $a_{n+k} = a_n$ for all $n \\ge N$.\nIt is purely periodic if this holds with $N = 0$.\n\n## Remarks\n\nOne writes the repeating block with a bar, for example $\\sqrt{2} = [1; \\overline{2}]$ and $\\sqrt{3} = [1; \\overline{1, 2}]$.\nBy theorems of Euler and Lagrange, a real number has a periodic continued fraction if and only if it is a quadratic irrational. That is, it is an irrational root of a quadratic equation with integer coefficients.\nThe period of $\\sqrt{d}$ gives the [[fundamental solution of Pell's equation|0KCYJSB9FW]].\n\n## Explanations\n\n### L1 human\n\nA periodic continued fraction repeats the same block of terms forever.\nThe golden ratio $[1; 1, 1, 1, \\dots]$ is the simplest example.\n",
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