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  "body": "\n## Problem\n\nLet $k \\ge 2$ be an [[integer|0C4JS9RF5M]].\nLet $g(k)$ be the least $s$ such that every positive integer is a sum of at most $s$ $k$-th powers of positive integers.\nLet $G(k)$ be the least integer $s$ such that every sufficiently large integer is a sum of at most $s$ $k$-th powers of positive integers.\nWaring's problem asks:\n\n1. whether $g(k)$ exists for every $k \\ge 2$;\n2. what the values of $g(k)$ and $G(k)$ are.\n\n## Remarks\n\nHilbert proved in 1909 that $g(k)$ exists for every $k$ (the Hilbert–Waring theorem).\nBy definition $G(k) \\le g(k)$.\n[[Lagrange's four-square theorem|0KYYSMEVHH]] gives $g(2) = 4$, and $G(2) = 4$ because no integer $n \\equiv 7 \\pmod 8$ is a sum of three squares.\nKnown values include $g(3) = 9$ and $g(4) = 19$; the numbers $23$ and $79$ need $9$ cubes and $19$ fourth powers.\nThe lower bound $g(k) \\ge 2^k + \\floor{(3/2)^k} - 2$ comes from the integer $n = 2^k \\floor{(3/2)^k} - 1$, which is less than $3^k$ and so must be written with the powers $1^k$ and $2^k$ only.\nEquality is known for all $k$ outside a possible finite set, and the possible exceptions have been excluded far beyond any small $k$.\nFor $G(k)$ much less is known: $G(4) = 16$ (Davenport), and $4 \\le G(3) \\le 7$, the upper bound being due to Linnik.\nThe [[Hardy–Littlewood circle method|ZQFBEX62D0]] gives an asymptotic formula for the number of representations when the number of powers is large compared with $k$.\n\n## Explanations\n\n### L1 human\n\nEvery positive integer is a sum of four squares, for example $7 = 4 + 1 + 1 + 1$.\nWaring asked whether the same is true for cubes, fourth powers and so on, with some fixed number of terms.\nThe answer is yes: for instance, every positive integer is a sum of at most nine cubes.\nThe hard part is to find the smallest number of terms that always suffices, or that suffices for all large numbers.\n",
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      "posed_by": "Edward Waring"
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