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Waring’s problem - mklabgr - 07-28-2026
Waring’s problem
Summary
Proposed by English mathematician Edward Waring in 1770, Waring's problem is a fundamental question in number theory asking whether, for every positive integer $k$, there exists a corresponding minimum integer $g(k)$ such that every natural number can be expressed as the sum of at most $g(k)$ natural numbers raised to the $k$-th power. For example, every natural number is the sum of at most 4 squares ($g(2) = 4$), 9 cubes ($g(3) = 9$), or 19 fourth powers ($g(4) = 19$).
David Hilbert affirmatively proved the existence of such a finite limit for every power $k$ in 1909—a result known as the Hilbert–Waring theorem—and modern research continues to explore exact formulas for $g(k)$ as well as $G(k)$, which measures the maximum number of $k$-th powers required to express all sufficiently large integers.
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