Lagrange's Four Square Theorem/Historical Note
Historical Note on Lagrange's Four Square Theorem
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.
Leonhard Paul Euler investigated the problem, discussing it with Christian Goldbach in around $1730$, publishing a proof of a weaker version around that time.
It was finally proved by Joseph Louis Lagrange in $1770$. The proof was published in $1772$.
Euler subsequently submitted a proof of his own in $1772$, which was finally published in $1780$.
Sources
- 1772: J.-L. Lagrange: Démonstration d'un théorème d'arithmétique (Nouveaux mémoires de l'Académie royale des sciences et belles-lettres de Berlin, Année 1770 pp. 123 – 133)
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $4$
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3$: Appendix $\text A$: Euler
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $4$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Lagrange's theorem: 1. (J.L. Lagrange, 1772)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Lagrange's theorem: 1. (J.L. Lagrange, 1772)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Waring's problem
- 2014: Jenny Boucard: Lagrange and the four-square theorem (Lettera Matematica Vol. 2: pp. 59 – 66)