Hilbert-Waring Theorem/Particular Cases/Mistake

Source Work

1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.):

Waring's problem


Mistake

The first part, on cubes, was proved by Hilbert in $1909$. The second part is still unsolved.


Correction

The first part, that is:

Every positive integer can be expressed as the sum of at most $9$ positive cubes

was in fact resolved by Arthur Josef Alwin Wieferich and Aubrey John Kempner between $1909$ and $1921$.


The second part, that is:

Every positive integer can be expressed as the sum of at most $19$ powers of $4$

had been resolved in $1986$ by Ramachandran Balasubramanian, Jean-Marc Deshouillers and François Dress, some $12$ years before the publication of this edition of this dictionary.


What Hilbert showed in $1909$ was that for every $n$ there exists a $\map g n$ such that every positive integer can be expressed by no more than $\map g n$ positive integers raised to the power of $n$.


These errors had both been corrected by the time of the publication of the $4$th edition.


Sources

  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Waring's problem
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