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  "body": "\n## Definition\n\nA *binary quadratic form* (with integer coefficients) is a polynomial\n$$f(x, y) = ax^2 + bxy + cy^2$$\nin two variables $x, y$, with coefficients $a, b, c$ in the [[integers|0C4JS9RF5M]].\nIts *discriminant* is $D = b^2 - 4ac$.\nThe form is *primitive* if the [[greatest common divisor|0C8WYW4KYQ]] of $a$, $b$, $c$ is $1$.\nTwo forms $f$ and $g$ are *properly equivalent* if $g(x, y) = f(\\alpha x + \\beta y, \\gamma x + \\delta y)$ for some matrix $\\begin{pmatrix} \\alpha \u0026 \\beta \\\\ \\gamma \u0026 \\delta \\end{pmatrix}$ in the [[special linear group|0V9PMN0RBH]] $\\mathrm{SL}_2(\\Z)$.\n\n## Remarks\n\nProperly equivalent forms have the same discriminant and represent the same integers.\nThe discriminant always satisfies $D \\equiv 0$ or $1 \\pmod 4$.\nThe form is positive definite if and only if $D \u003c 0$ and $a \u003e 0$.\nSome authors write the form as $ax^2 + 2bxy + cy^2$ and require $a, b, c$ to be integers; this is Gauss's convention.\n\nExample: $x^2 + y^2$ has $a = 1$, $b = 0$, $c = 1$, so it is a primitive binary quadratic form of discriminant $-4$.\n\nNon-example: $x^2 + y^2 + z^2$ is a quadratic form with integer coefficients, but it has three variables, so it is not binary.\n",
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