> Second, we have
> **********
> the minimum $r$ such that b^r=1 (mod n).
> **********
> Note that this second idea is never (to my knowledge) called an
> index.
I can give two counter-examples to this "never":
1) The cause of my research on this topic, Avinoam Mann, who wrote
to me:
***the "order" is sometimes referred to in number theory books,
especially older ones, as "index".***
2) Borel himself, in his "Les nombres premiers" (PUF, Paris 1953),
p. 63 after repeating his 1895 phrasing, adds:
************
nous conviendrons de dire aussi que r est l'indice de b
************
This is interesting news; as I said, I have never seen that usage.
Gauss definitely doesn't use it. But I now have a guess of what
might have happened. The number is sometimes called the exponent
to which b belongs, and I seem to remember that at one time some
writers on algebra used "index" where we would use "exponent".
So that is a possible explanation. Not knowing any examples of
the usage in this context (Borel's book isn't in our library),
I can't check for myself; perhaps someone else will know where
to look.
William C. Waterhouse
Penn State