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  "body": "\n## Statement\n\nFor each positive integer $k$, there is a [[Hadamard matrix|DM59GHSMAJ]] of order $4k$.\n\n## Remarks\n\nA Hadamard matrix of order $n \u003e 2$ can exist only when $n$ is a multiple of $4$, so the conjecture says that this necessary condition is also sufficient.\nThe conjecture is open.\nSylvester's construction by [[Kronecker products|0JK6M18QZH]] gives all orders $2^m$.\nPaley's construction uses [[quadratic residues|0SQDB87Z3T]] in a [[finite field|08RTDHX1SK]] $\\F_q$ and gives order $q + 1$ when $q \\equiv 3 \\pmod 4$ is a prime power.\nCombining such constructions covers many multiples of $4$; the smallest multiple of $4$ for which no Hadamard matrix is known is $668$.\n\n## Explanations\n\n### L1 human\n\nA Hadamard matrix is a square table of plus and minus signs in which any two rows agree in exactly half of the places.\nSimple counting shows that the size must be a multiple of $4$, apart from sizes $1$ and $2$.\nThe conjecture says that every multiple of $4$ really is possible.\n",
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    "history": {
      "first_stated": "1933",
      "posed_by": "Raymond Paley"
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    "kind": "conjecture",
    "msc": [
      "05B20"
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    "name": "Hadamard conjecture",
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        "url": "https://en.wikipedia.org/wiki/Hadamard_matrix"
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  "kind": "conjecture",
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  "markdown": "https://math2.tamnd.com/node/6X9S10E3M2.md",
  "msc": [
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  "name": "Hadamard conjecture",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/6X9/6X9S10E3M2.md",
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  "tag": "6X9S10E3M2",
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  "url": "https://math2.tamnd.com/wiki/hadamard-conjecture/",
  "writing": {
    "tag": "6X9S10E3M2",
    "kind": "conjecture",
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