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  "body": "\n## Statement\n\nLet $d$ be a positive squarefree integer.\nThe imaginary [[quadratic field|7SV4WA92SM]] $\\Q(\\sqrt{-d})$ has [[class number|KTKK9B1X41]] $1$ if and only if\n$$d \\in \\set{1, 2, 3, 7, 11, 19, 43, 67, 163}.$$\n\n## Remarks\n\nEquivalently, the fundamental discriminants $D \u003c 0$ with class number $1$ are $-3, -4, -7, -8, -11, -19, -43, -67, -163$.\nClass number $1$ means that the ring of integers is a principal ideal domain, or, equivalently for rings of integers, a [[unique factorisation domain|0W4X9G6263]].\nGauss's work on [[binary quadratic forms|SBDSYA037Q]] led him to expect this list; the theorem settles the case $h = 1$ of the [[class number problem|KTGAS9DGRX]].\nHeegner gave a proof using modular functions and [[complex multiplication|D3HEJTVY7G]], which was at first not accepted.\nBaker, with linear forms in logarithms, and Stark, with a method close to Heegner's, later gave complete proofs.\n\nExample: $\\Q(\\sqrt{-163})$ has class number $1$.\nA related fact is that $n^2 + n + 41$ is prime for $n = 0, 1, \\ldots, 39$; this works because $163 = 4 \\cdot 41 - 1$.\nBy contrast $\\Q(\\sqrt{-5})$ has class number $2$, as $6 = 2 \\cdot 3 = (1 + \\sqrt{-5})(1 - \\sqrt{-5})$ shows.\n\n## Explanations\n\n### L1 human\n\nIn most rings of algebraic integers, numbers can factor into irreducibles in several different ways.\nFor imaginary quadratic fields, the theorem gives the complete list of the nine fields where factorisation is unique.\nThe largest one comes from $\\sqrt{-163}$.\n\n### L2 human\n\nThe main difficulty is to rule out one possible extra field with a huge discriminant.\nLower bounds for class numbers of the form $h(-d) \\to \\infty$ were known, but they were not effective, so they could not exclude a single exception.\nThe theorem is closely related to the near-integer $e^{\\pi \\sqrt{163}}$, which comes from the $j$-invariant at a point of complex multiplication.\n",
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