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  "body": "\n## Statement\n\nFor a nonzero polynomial $f \\in \\Z[x]$, let $M(f)$ be its [[Mahler measure|QVPYBPRHRC]].\nThere is a constant $c \u003e 1$ such that $M(f) \\ge c$ for every $f \\in \\Z[x]$ with $M(f) \u003e 1$.\n\n## Remarks\n\nThe condition $M(f) \u003e 1$ excludes exactly the polynomials $\\pm x^k \\Phi_{n_1} \\cdots \\Phi_{n_r}$ built from [[cyclotomic polynomials|08KEYFAH66]], which have $M(f) = 1$.\nThe problem goes back to D. H. Lehmer, who asked it in 1933.\nThe conjecture is open.\nThe smallest known value of $M(f) \u003e 1$ is that of Lehmer's polynomial $x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1$, about $1.17628$.\nSmyth proved the conjecture for nonreciprocal polynomials.\nIf $f \\in \\Z[x]$ is irreducible of degree $d$, $f \\ne \\pm x$, and $x^d f(1/x) \\ne \\pm f(x)$, then $M(f) \\ge M(x^3 - x - 1) \\approx 1.3247$.\nDobrowolski proved a lower bound of the form $M(f) \\ge 1 + c \\left( \\frac{\\log \\log d}{\\log d} \\right)^3$ for an irreducible noncyclotomic $f$ of degree $d$, with an absolute constant $c \u003e 0$.\n\n## Explanations\n\n### L1 human\n\nThe Mahler measure of an integer polynomial is at least $1$, and it equals $1$ only for products of cyclotomic polynomials and powers of $x$.\nThe conjecture says that there is a gap: the next possible values cannot come arbitrarily close to $1$.\nThe best candidate for the smallest value above $1$ has been known for almost a century.\n",
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    "history": {
      "first_stated": "1933",
      "posed_by": "Derrick Henry Lehmer"
    },
    "kind": "conjecture",
    "msc": [
      "11R06",
      "11C08"
    ],
    "name": "Lehmer's conjecture",
    "same_as": {
      "wikidata": "Q6518929",
      "wikipedia": "Lehmer's conjecture"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lehmer%27s_conjecture"
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    "tag": "NDRRH2YD2C",
    "uses": [
      "QVPYBPRHRC",
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  "kind": "conjecture",
  "link": "https://math2.tamnd.com/node/NDRRH2YD2C/",
  "links": [
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      "target": "08KEYFAH66"
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  "markdown": "https://math2.tamnd.com/node/NDRRH2YD2C.md",
  "msc": [
    "11R06",
    "11C08"
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  "name": "Lehmer's conjecture",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/NDR/NDRRH2YD2C.md",
  "status": null,
  "tag": "NDRRH2YD2C",
  "uri": "m2://NDRRH2YD2C",
  "url": "https://math2.tamnd.com/wiki/lehmers-conjecture/",
  "writing": {
    "tag": "NDRRH2YD2C",
    "kind": "conjecture",
    "level": "W0",
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        "what": "a source with a locator",
        "have": 0,
        "want": 1
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