Hilbert-Waring Theorem/Particular Cases/2
Particular Case of the Hilbert-Waring Theorem: $k = 2$
The Hilbert-Waring Theorem states that:
The case where $k = 2$ is proved by Lagrange's Four Square Theorem:
- $\map g 2 = 4$
Source of Name
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 $1770$.