In Cauchy's Oeuvres, t. 3, Paris (1911) p. 84, (from Me/moires Acad. Sc.
Paris t. 17-1840), I find:
*Nous disons, avec Euler (...) que $r$ est "racine primitive" (...)*
or:
*Following Euler, we say $r$ is "primitive root"*
Which situates, at least, to Cauchy the opinion that Euler has some rights
to "primitive root"'s paternity. And, thanks to Heinz Lueneburg [Mon 23 Aug
1999], we now know about this paternity's consistency.
Then, in Legendre's Essai sur la the/orie des nombres, Paris (an VI-1798),
p. 413-414, we read (my translation):
***************
Theorem (348): Being proposed the equation (x^n-1)/a = e, where $a$ is prime,
and $n$ divides a-1.
(...)
2nd. If $t$ is a value for $x$, t^m will be too, whatever the exponent $m$ is.
3rd. If the number $t$ is such that t^(n/v)-1 is not divisible by $a$, $v$
being a prime divisor of $n$, then the formula x = t^m will contain all the
solutions of the proposed equation, and they will be 1, t, t^2, ... t^(n-1),
or the rest of these quantities divided by $a$.
***************
I personally find this Legendre's proposition a quite acceptable definition
for both notions of "minimum exponent" and "primitive root". So to say, with
Ed Sandifer's permission [Sun 22 Aug 1999]:
Even Gauss couldn't do everything.
Udai Venedem
venedem@wanadoo.fr
http://perso.wanadoo.fr/alta.mathematica/
(new catalogue, and thematic pages - one in particular, on number theory -
in French)