{
  "aliases": [
    "coin-problem",
    "frobenius-number"
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  "backlinks": null,
  "body": "\n## Statement\n\nLet $a$ and $b$ be [[coprime|0GFKP8F8AG]] integers with $a \u003e 1$ and $b \u003e 1$.\nThen $ab - a - b$ is not of the form $ax + by$ with natural numbers $x$ and $y$.\nEvery integer $n \u003e ab - a - b$ is of this form.\n\n## Remarks\n\nThe number $ab - a - b$ is the Frobenius number of $a$ and $b$: the largest amount that one cannot pay with coins of values $a$ and $b$.\nFor example, with coins of $3$ and $5$, the amount $7$ is impossible, but $8$, $9$, $10$ and all larger amounts are possible.\nExactly $(a - 1)(b - 1)/2$ positive integers are not of the form $ax + by$.\nFor three or more coin values, no such simple formula is known.\n\n## History\n\nJames Joseph Sylvester found the formula for two coins in 1882.\n\n## Explanations\n\n### L1 human\n\nWith coins of values $a$ and $b$, you can pay all large amounts when $a$ and $b$ are coprime.\nThe theorem gives the largest amount that you cannot pay: $ab - a - b$.\n",
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    "name": "Frobenius coin problem",
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      "wikidata": "Q2295746",
      "wikipedia": "Coin problem"
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        "locator": "n = 2",
        "url": "https://en.wikipedia.org/wiki/Coin_problem"
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  "msc": [
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  "name": "Frobenius coin problem",
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  "role": "theorem",
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