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  "body": "\n## Summary\n\nAn integral quadratic form is a [[quadratic form|0RBQS1JJCR]] $\\sum_{i \\le j} a_{ij} x_i x_j$ whose coefficients $a_{ij}$ are integers.\nThe main questions are arithmetic: which integers the form represents, in how many ways, and when two forms are equivalent under integral changes of variables.\nClassical examples are sums of squares, such as $x_1^2 + x_2^2 + x_3^2 + x_4^2$.\nIntegral quadratic forms correspond to [[lattices|WN9V7WSJKF]] with an integer-valued quadratic function on them.\n\n## Explanations\n\n### L1 human\n\nTake a quadratic polynomial such as $x^2 + xy + y^2$ or $x^2 + y^2 + z^2$ with whole number coefficients.\nPlugging in whole numbers gives whole numbers, and one asks which numbers come out.\nLagrange proved that every nonnegative integer is a sum of four squares, so the form $x_1^2 + x_2^2 + x_3^2 + x_4^2$ takes every such value.\nThe form $x^2 + y^2 + z^2$ misses $7$, since $7$ is not a sum of three squares.\n",
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