Indeed that IS false - though the Jacobi symbol has the
corresponding property.
>> 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."
... and he should have said that their statements should be
modified to interpret R as meaning that the yet-to-be-defined
Jacobi symbol is 1.
William Waterhouse's remarks ...
> As one would guess, of course, there is no error. The notation
> is explained just after the statement of quadratic reciprocity
> in article 131:
>
> "To indicate our reasoning as briefly as possible, we will
> denote prime numbers of the form 4n+1 by the letters a, a', a'', etc.
> and prime numbers of the form 4n+3 by the letters b, b', b'', etc.;
> any numbers of the form 4n+1 will be denoted by A, A', A'', etc;
> any numbers of the form 4n+3 by B, B', B'', etc...."
>
> The capital letters introduced there just aren't needed until art. 132.
> The notation continues in effect through art. 134.
... are contradicted by the explicit assertion that the numbers need
not necessarily be prime in the particular theorem under discussion.
I don't recall observing this mistake when I read the Disquisitiones
(emulating Eistenstein, I used to sleep with it under my pillow) in
the German version, although I DO remember thinking that Gauss almost
certainly discovered the Jacobi symbol (so to speak). But I'm not
perfect, though it pains me to admit it, and so might easily have failed
to spot it.
Which edition did you find it in, Franz?
John Conway