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do we find, and where, in Euler 1) the notion of "minimum power"
(in French: "exposant minimal") and 2) the possibility of expressing
any residue by a power of a primitive root?
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I quote Andre Weil's book, "Number Theory, an Approach Through History", parts
of pages 194 and 195:
Discussing E 262 (1775, Opera Omnia I-2 pp 493-518)
"Incidentally, he also notes [in a Scholion on page 510] that phi(n) is not in
general the smallest exponent n such that N divides x^n^-1 for all x prime to
N. ..."
Discussing the Tractatus (Euler's manuscript on Number Theory, unpublished
until the mid 1800's, section 255)
"... more generally, he uses it [the method of finite differences] to show that
if n is less than p-1, p cannot divide a^n-1 for all a prime to p.
"Here he narrowly missed an opportunity for showing the existence of
'primitive roots' modulo an arbitrary prime. ..."
If these answer your questions, and if Weil is correct, it would seem
that Euler deserves little credit for your question (1), since his manuscript
wasn't published until 50 years after Gauss published it, and he deserves no
credit for your question (2).
Even Euler couldn't do everything.
Ed Sandifer