> 3. A.Weil's masterpiece "Number Theory" has an excellent discussion
> of Diophantus on pp.24-34. His accounts of the work of Fermat, Euler,
> Lagrange, and Legendre in number theory are just as impressive.
> There is much in this book on the history of congruences.
Can someone steer me towards a specific discussion of the topic of
congruence arguments in Weil? I just got a copy, and I don't see the
passages that Abe is referring to. (The book's full title is "Number
Theory, an approach through history: From Hammurapi to Legendre".)
The only discussion of congruence arguments prior to Fermat that I can find
discussed in the book (leaving aside the special case of parity arguments,
which go back at least as far as ancient Greece) is on pp. 30-31, where
Weil discusses Diophantus's few observations that are genuinely "Diophantine"
in the modern sense (e.g., that 15 cannot be written as a sum of two squares
in the rationals); after mentioning that Fermat proved such things with
congruence arguments mod 4 and 8, Weil says "Such arguments are not found
anywhere in Diophantus, but they cannot be said to be alien to the spirit
of Greek arithmetic. Short of a chance discovery of hitherto unknown texts,
the questions thus raised are unanswerable," and the next mention of
congruence arguments that I see in the book concerns Fermat.
Is there really no history of explicit congruence arguments in between the
time of Diophantus and the time of Fermat (other than Leonardo of Pisa, who
in his Liber Quadratorum uses congruences mod 3)?
Also: Does anyone know if any publisher has plans to reissue Weil's book?
Last time I checked, it was out of print; I'd love to own a copy.
Abe also writes:
> 5. What you refer to as "negative results" is usually stated as a
> restriction ("porism") on the numbers appearing in a "positive result."
Can someone give me an explanation of exactly what a "porism" is? I believe
that the standard usage of the term is a bit broader than restrictions-on-
positive-results; e.g., Weil (pp. 11-12) says that the identity
2 2 2 2 2 2
(x + y ) (z + t ) = (xz + yt) + (xt - yz)
may be one of Diophantus' lost porisms. So, what sort of theorems get called
porisms?
Jim Propp