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)


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 Fp. 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.

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