E242  Demonstratio theorematis Fermatiani omnem numerum sive integrum sive fractum esse summam quatuor pauciorumve quadratorum
(Proof of a theorem of Fermat that every number whether whole or fraction is the sum of four or fewer squares)
Summary:
This paper contains the beginnings of group theory, showing that quadratic residues of a prime p form a subgroup of index 2 of the multiplicative group F_{p}. Euler ends up with a result about rational squares rather than about integer squares, so it falls just short of its mark. Since there is no title page for this paper, Eneström suggests that it is a only a continuation of E241.
According to C. G. J. Jacobi, a treatise with the title given above was read to the Berlin Academy on June 17, 1751.
Publication:

Originally published in Novi Commentarii academiae scientiarum Petropolitanae 5, 1760, pp. 1358

Opera Omnia: Series 1, Volume 2, pp. 338  372
 According to Jacobi, the manuscript of the printed treatise
can be found in the archive of the Berlin Academy.
 Reprinted in Commentat. arithm. 1, 1849, pp. 215233 [E242a]
 A handwritten French translation of this treatise can be found in the library of the observatory in Uccle, near Brussels.
Documents Available:
 Original publication: E242
 English translation (Paul Bialek): E242
 The Euler Archive attempts to monitor current scholarship for articles and books that may be of interest to Euler Scholars. Selected references we have found that discuss or cite E242 include:
 Pieper H., “On Euler contributions to the 4squares theorem.” Historia Mathematica, 20 (1), pp. 1218 (Feb 1993).
 Rudio
 Weil A., Number Theory
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