Hilbert-Waring Theorem/Particular Cases/5

Particular Case of the Hilbert-Waring Theorem: $k = 5$

The Hilbert-Waring Theorem states that:

For each $k \in \Z: k \ge 2$, there exists a positive integer $\map g k$ such that every positive integer can be expressed as a sum of at most $\map g k$ positive $k$th powers.

The case where $k = 5$ is:

Every positive integer can be expressed as the sum of at most $37$ positive fifth powers.

That is:

$\map g 5 = 37$

Proof

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Also see

• Largest Number not Expressible as Sum of Less than 37 Positive Fifth Powers

• Largest Number not Expressible as Sum of Less than 32 Positive Fifth Powers

Historical Note

This result was demonstrated by Jingrun Chen in $1964$.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $37$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $37$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Waring's problem