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  "body": "\n## Definition\n\nLet $d$ be a positive integer that is not a perfect square.\nPell's equation for $d$ is the [[Diophantine equation|08B1RTW7VZ]] $x^2 - d y^2 = 1$, with unknowns $x, y \\in \\mathbb{Z}$.\nA solution is a pair $(x, y)$ of integers that satisfies the equation.\n\n## Remarks\n\nThe pairs $(\\pm 1, 0)$ are the trivial solutions.\nFor $d = 2$, the pair $(3, 2)$ is a solution, since $9 - 2 \\cdot 4 = 1$.\nA solution $(x, y)$ with $y \\neq 0$ gives the rational approximation $x / y$ of $\\sqrt{d}$.\nThe name comes from an error of Euler, who attributed a solution by William Brouncker to John Pell.\n\n## Explanations\n\n### L1 human\n\nPell's equation asks for integers $x$ and $y$ such that $x^2$ is exactly one more than $d y^2$.\nFor $d = 2$, the solutions $(3, 2)$, $(17, 12)$ and $(577, 408)$ give fractions very close to $\\sqrt{2}$.\n",
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