As this discussion took place while I was not on the list,
here's a late contribution:
As I indicate on page 1, Chapter 1, of my manuscript on
the evolution of reciprocity laws from Euler to Artin
( http://www.rzuser.uni-heidelberg.de/~hb3/rec.html ),
according to J. Hofmann,
"Uber die zahlentheoretischen Methoden Fermats und
Eulers, ihre Zusammenh"ange und ihre Bedeutung},
Archive for History of Exact Sciences 1 (1960/62), 122--159,
Fermat's 2-squares theorem first appeared in a book of Simon Stevin.
As others have already remarked, apparently nobody doubts that
Fermat was the first who could prove this, and that the first
published proof was Euler's.
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I have put a few things on the more recent history of
number theory (a lecture by Lipschitz, an unpublished
article by A. Scholz, and the introduction to the English
edition of Hilbert's Zahlbericht) on
http://www.rzuser.uni-heidelberg.de/~hb3/hist.html
and while I'm at it, some very interesting articles are
to be found on Roquette's homepage
http://www.rzuser.uni-heidelberg.de/~ci3/manu.html
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bye,
franz