Yesterday, I forgot to mention that the name "primitive root" is in Euler's
paper I quoted already yesterday (p. 88/89 of the original, p. 244 of the
opera).
Demonstrationes circa residua ex divisione potestatum per numeros
primos resultantia. Novi commentarii academiae scientiarum Petropolitanae 18
(1773), 85-135, 1774, Opera omnia II, 240-281
There it reads: Huiusmodi radices progressionis geometricae, quae series
residuorum completas producunt, {\it primitivas} appellabo.
From Udai Venedem's message of today:
>
> 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 $a$.
> ***************
Here primitive roots are characterized. Does Legendre also prove the existence
of primitive roots?
By the way, Euler does not seem to have a name for the multiplicative order of
an element modulo a prime. However, he sees clearly the importance of that
number. The papers to look for such a name are the two papers where he gave his
third proof of the little Fermat theorem and its generalization. I could not
find a name for this number in these papers. The two papers are:
Theoremata circa residua ex divisione potestatum relicta. Novi commentarii
academiae scientiarum Petropolitanae 7 (1758/9), 49-82, 1761. Opera omnia I,
493-518
Theoremata arithmetica nova methodo demonstrata. Novi commentarii academiae
scientiarum Petropolitanae 8 (1760/1), 74-104, 1763. Opera omnia I, 531-555
Heinz Lueneburg