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  "body": "\n## Statement\n\nLet $d$ be a positive integer that is not a perfect square.\nThen [[Pell's equation|0AF3DANJ76]] $x^2 - d y^2 = 1$ has a solution in integers with $y \\neq 0$.\nIn fact, it has infinitely many solutions in integers.\n\n## Remarks\n\nOne proof uses [[Dirichlet's approximation theorem|04JAY9DGB7]] to find infinitely many $x/y$ with $|x^2 - d y^2|$ bounded.\nAnother proof uses the [[periodic continued fraction|0XKJPFS2W0]] of $\\sqrt{d}$.\nThe smallest solution can be very large. For $d = 61$, it is $(1766319049, 226153980)$.\n\n## History\n\nIndian mathematicians, such as Brahmagupta, studied the equation many centuries before European mathematicians.\nLagrange proved that it always has infinitely many solutions.\n\n## Explanations\n\n### L1 human\n\nFor each non-square $d$, there are integers $x$ and $y \\neq 0$ with $x^2 - d y^2 = 1$.\nThey may be hard to find by trial: for $d = 61$, the smallest $x$ has ten digits.\n",
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