| At 10:24 AM 12/01/1999 -0500, Jim Propp <propp@math.mit.edu> wrote:
|
| | Julio (and invited eavesdroppers),
| |
| | I've just read your July 9
|
| January 9
Yes, that is what I meant. (I also meant to write "1999" instead of "1998"
on the three checks that I just wrote, but that's another matter.) :-)
Anyway, I think Pepin is being unfair to Euler when he writes
Next, having recognized that the only solution
for which $y$ equals +1 (or -1) corresponds to the values p=q=+1/2
(or both -1/2), that render x=+2 (or -2), y=+1 (or -1), Euler concludes
that 4 is the only square which answers the question.
Do you agree?
Also, in view of
3. We cannot regard any of the two Fermat's theorems as entirely
proven, which is the object of the first two questions of Chapter
XII, as long as we have not justified the use of the preceding
method for the two formulae x^2 + y^2, x^2 + 2y^2. But, this is
what was neither worked out by Euler nor by Legendre. On this issue,
Legendre contented himself with quoting Euler's demonstrations.
it would be interesting to know what Legendre said, and whether he treated
Euler's partial analysis as if it were (or claimed to be) a full solution.
Thanks, Julio, for your industrious work on translating these passages!
I hope others are benefitting from them as much as I have.
Jim Propp
Department of Mathematics
University of Wisconsin