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  "body": "\n## Statement\n\nFor a positive integer $n$, let $\\operatorname{rad}(n)$ be the product of the distinct [[primes|01826KMG2H]] that divide $n$.\nFor every real number $\\varepsilon \u003e 0$, there are only finitely many triples $(a, b, c)$ of [[coprime|0VZ48N083F]] positive integers with $a + b = c$ and\n$$c \u003e \\operatorname{rad}(abc)^{1 + \\varepsilon}.$$\n\n## Remarks\n\nThe conjecture was formulated by Joseph Oesterlé and David Masser in 1985.\nShinichi Mochizuki announced a proof in 2012; it is not accepted by most experts, and the conjecture is generally treated as open.\nThe analogue for polynomials is a theorem, the [[Mason–Stothers theorem|ZWPWF6NJC3]].\nThe $\\varepsilon$ cannot be removed.\nFor each $k \\ge 1$, take $a = 1$, $c = 2^{6k}$ and $b = c - 1$; since $2^6 = 64 \\equiv 1 \\pmod 9$, the number $9$ divides $b$, so $\\operatorname{rad}(abc) \\le 2 \\cdot b/3 \u003c c$.\nFor example, $1 + 8 = 9$ has $\\operatorname{rad}(1 \\cdot 8 \\cdot 9) = 6 \u003c 9$, and $5 + 27 = 32$ has radical $30 \u003c 32$.\nThe conjecture would imply many results, for instance that the Fermat equation $x^n + y^n = z^n$ has no solutions in positive integers for all large $n$.\n\n## Explanations\n\n### L1 human\n\nThe radical of a number keeps each prime factor only once, so $\\operatorname{rad}(72) = \\operatorname{rad}(2^3 \\cdot 3^2) = 6$.\nIf $a + b = c$ and the three numbers share no factor, then usually $c$ is smaller than the radical of $abc$.\nThe conjecture says that $c$ can exceed the radical only by a little, and even this happens only rarely in a precise sense.\nIt expresses a tension between adding numbers and multiplying primes, and it has a remarkable list of consequences.\n",
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      "first_stated": "1985",
      "posed_by": "Joseph Oesterlé and David Masser"
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    "kind": "conjecture",
    "msc": [
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    "name": "abc conjecture",
    "same_as": {
      "wikidata": "Q306393",
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  "link": "https://math2.tamnd.com/node/G7JH2NXPC0/",
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  "msc": [
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  "name": "abc conjecture",
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