Hilbert-Waring Theorem/Particular Cases/2

From ProofWiki
Jump to navigation Jump to search

Particular Case of the Hilbert-Waring Theorem: $k = 2$

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 $\map g k$ $k$th powers.

The case where $k = 2$ is proved by Lagrange's Four Square Theorem‎:

$g \left({2}\right) = 4$

That is, every positive integer can be expressed as the sum of at most $4$ squares.

Source of Name

This entry was named for David Hilbert and Edward Waring.

Historical Note

It is suggested by some sources that the result of Lagrange's Four Square Theorem was known, at least empirically, by Diophantus of Alexandria.

Some sources suggest that the theorem was originally stated formally by Pierre de Fermat.

However, it appears that Claude Gaspard Bachet de Méziriac published the results of his having tested it thoroughly up to $120$, and stated the theorem in his $1621$ translation of the Arithmetica of Diophantus.

Fermat read about it in his copy of that work, and studied it, but appears to have failed to find a proof, as no proof of his can be found.

Some sources claim that its first proof was by Leonhard Paul Euler, but this is questionable.

It was finally proved by Joseph Louis Lagrange in $1770$.