During a week or two, Avinoam Mann and I had an 'aparte' on the subject
of number theory (it had started from Lucas and Proth). Incidentally
(as always in the matter of history), we found a difference in the use
of the term "index" by him ("group theorist") or by me (referring to
"older" doctrines).
In the following, let us suppose $n$ to be prime.
What group theorists call "index" of $b$, is the minimum $r$ such that
b^r=1 (mod n). And Lucas named it "le gaussien de $b$ pour le module $n$"
(Theorie des nombres, Paris 1891, p. 439). Other authors just did not
name it, and one of them only said:
*the number $b$ belongs to the power $r$ with respect to the prime $n$*
(as quoted from Tannery-Borel-Drach, Theorie des nombres et de l'algebre,
Paris 1895, p.51). But, as late as 1922 (Kraitchik, Theorie des nombres,
Paris, p. 115), we find the following different meaning of "index":
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If $a$ is primitive root of $n$, the congruence a^x=b (mod p) is always
possible, except if b = 0 (mod p), and has only one solution less than
p-1. We call $x$ the index of $b$ (mod n) in the system of base $a$.
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It would be interesting enough to determine the first appearances of
the term "index" in both acceptations (for the "old fashion" one, I
propose Jacobi's Canon arithmeticus, Berlin 1839). We can exhibit the
equation: "new fashion index" = (n-1)/"old fashion index".
Udai Venedem
venedem@wanadoo.fr
http://perso.wanadoo.fr/alta.mathematica/