while reading my newly acquired Disquisitiones arithmeticae
I stumbled across something in Gauss's first proof of the
quadratic reciprocity law that seems odd to me: after stating
the theorem in article 131, he deduces the reciprocity law
for the Jacobi symbol in 133 (in different language: Gauss
didn't have the Jacobi symbol). Nothing surprising here,
but in article 132 he states some theorems that, if I
understand them correctly, are false. The first one reads
9. a R A ===> A R a,
where m R n is short for "there exists an integer x such that
m \equiv x^2 \bmod n", and where a and A are _integers_
(not necessarily prime) of the form 4r+1.
Now clearly 21 R 5 since 21 = 1^2 mod 5, but 5 N 21
(since 5 N 3) so we have a counterexample. The proof that
Gauss gives for 9 is correct for _prime_ values of a and
holds for arbitrary integers A.
Although Gauss, e.g. in article 108, uses "number" in the
sense of "prime number", in article 132 he says expressis
verbis that "This table [in art. 131] includes all the
cases when two prime numbers are compared; what follows
pertains to any numbers, but the demonstration of these
is less obvious."
Now my question: has this error of Gauss been observed
before? I can hardly believe that it has been overlooked,
but I checked that Smith (in his history of # theory) does
not mention it, and Dickson's history has nothing on
reciprocity. Moreover Dirichlet in his revision of Gauss's
first proof doesn't say a word about it, although _he_
couldn't possibly have overlooked this.
franz
P.S. I've put the list of published proofs (known to me)
of the quadratic reciprocity law online at
http://www.rzuser.uni-heidelberg.de/~hb3/rchrono.html
Three of them are by Mertens, by the way. References
will be added as soon as I find the time. My complete
bibliography on reciprocity laws can be found at
http://www.rzuser.uni-heidelberg.de/~hb3/recbib.html
I'm working on adding links to the reviewing journals
FdM, Zbl and MR.