{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0TA7WT68RK"
    }
  ],
  "body": "\n## Definition\n\nLet $a$ be an integer with $a \u003e 1$.\nA Fermat pseudoprime to the base $a$ is a [[composite number|0N3ADXHNTQ]] $n$ such that $a^{n-1} \\equiv 1 \\pmod n$.\n\n## Remarks\n\n[[Fermat's little theorem|0M25PRN3WQ]] says that each prime $n$ with $n \\nmid a$ satisfies $a^{n-1} \\equiv 1 \\pmod n$. A Fermat pseudoprime is a composite number that passes the same test.\nThe smallest Fermat pseudoprime to the base $2$ is $341 = 11 \\cdot 31$.\nThe condition implies $\\gcd(a, n) = 1$.\nA composite number that is a Fermat pseudoprime to each base coprime to it is a [[Carmichael number|0TA7WT68RK]].\n\n## Explanations\n\n### L1 human\n\nThe [[Fermat primality test|0N5ZSZMYH6]] declares $n$ \"probably prime\" when $a^{n-1} \\equiv 1 \\pmod n$.\nA Fermat pseudoprime is a composite number that fools this test for the base $a$.\nFor example, $2^{340} \\equiv 1 \\pmod{341}$, but $341$ is not prime.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "defines": "0W93FJAC8F",
    "kind": "definition",
    "msc": [
      "11A51"
    ],
    "name": "Fermat pseudoprime",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Fermat_pseudoprime"
      }
    ],
    "tag": "02M2K4QH9P",
    "uses": [
      "0N3ADXHNTQ",
      "0TPW25CBN6",
      "0M25PRN3WQ"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/02M2K4QH9P/",
  "links": [
    {
      "field": "defines",
      "target": "0W93FJAC8F"
    },
    {
      "field": "uses",
      "target": "0N3ADXHNTQ"
    },
    {
      "field": "uses",
      "target": "0TPW25CBN6"
    },
    {
      "field": "uses",
      "target": "0M25PRN3WQ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/02M2K4QH9P.md",
  "msc": [
    "11A51"
  ],
  "name": "Fermat pseudoprime",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/02M/02M2K4QH9P.md",
  "status": null,
  "tag": "02M2K4QH9P",
  "uri": "m2://02M2K4QH9P",
  "url": "https://math2.tamnd.com/wiki/fermat-pseudoprime-definition/",
  "writing": {
    "tag": "02M2K4QH9P",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}