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  "body": "\n## Summary\n\nA binary quadratic form is a [[quadratic form|0RBQS1JJCR]] in two variables, $ax^2 + bxy + cy^2$, usually with integer coefficients.\nIts discriminant $b^2 - 4ac$ is unchanged by integral changes of variables of determinant $1$.\nGauss developed the theory of these forms: which integers they represent, how they fall into equivalence classes, and how the classes combine into a group.\nThis theory is equivalent to the arithmetic of ideals in [[quadratic fields|7SV4WA92SM]], and the number of classes is a [[class number|KTKK9B1X41]].\n\n## Explanations\n\n### L1 human\n\nThe forms $x^2 + y^2$ and $x^2 + 2y^2$ are binary quadratic forms.\nA classical question is which numbers they produce when $x$ and $y$ run through the integers.\nFermat found that an odd prime is of the form $x^2 + y^2$ exactly when it leaves remainder $1$ on division by $4$.\nChanging variables by an integer matrix of determinant $1$ does not change the set of values, so such forms are grouped into classes.\n",
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