>> "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."
I pointed out that the notation involved a convention introduced
just a bit earlier (art. 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...."
Now (Tue, 28 Sep 1999) John Conway <conway@math.Princeton.EDU>
maintains that the claim that this convention is in use is
"contradicted by the explicit assertion that the numbers need
not necessarily be prime in the particular theorem under
discussion."
I am not persuaded of this. There is no formal theorem-statement
involved; the words are just a transition sentence. Let me summarize
in a bit more detail what happens on the two pages in question.
In the first paragraph, Gauss "for brevity" introduces a
notational convention to show at a glance whether a number is
1 or 3 mod 4 and whether or not it is assumed to be prime. He then
states 8 different versions of the "fundamental theorem" purely in
symbols. He does not say in words that the numbers involved are
both primes, but his symbols mark them as such.
The next paragraph begins with the transitional sentence already
quoted. He then makes the "same" eight statements, except that now
the notation marks only the first of the two numbers as prime.
Then he says that the proofs of these all rest on the same ideas,
so he will only give one. He begins by pointing out how the class
of a number mod 4 is reflected in the parity of the number of
prime factors that are 3 mod 4.
Finally, he proves the first of the eight new statements. His
proof introduces the prime factors of the number not marked prime
and uses the results from the first eight; that is, the proof takes
for granted that the number marked as prime is indeed prime. He
then begins a discussion of what can be said when neither number
is assumed prime.
Clearly the mathematics here is correct and is part of a coherent
structure. If we are to say there is a mistake, it can only be
that the wording of the transitional sentence is too ambiguous.
The version quoted here was that of the English translation, but
there's no serious discrepancy from others. The German is
"Hierin sind alle Faelle, welche bei der Vergleichung
zweier Primzahlen vorkommen koennen, enthalten; das
Folgende bezieht sich auf beliebige Zahlen, doch liegen
die Beweise dafuer nicht so auf der Hand."
The original Latin is
"In his omnes casus, qui, duos numeros primos comparando,
occurrere possunt, continentur: quae sequuntur, ad numeros
quoscunque pertinent; sed horum demonstrationes minus sunt
obviae."
William C. Waterhouse
Penn State