Now I've read Dickson, but I'm still confused.
Fermat claimed, without proof (of course), that the only integer solution
is 25 = 27 - 2. Nothing confusing there. But now the confusion starts.
According to Dickson (volume II, pages 533-534), Euler, in his treatment
of Fermat's assertion (Algebra, II, chapter 12, Articles 187-195; or,
Opera Omnia, (1), I, 429-434), made axx+byy a cube by assuming that
x sqrt(a) + y sqrt(-b) is of the form (p sqrt(a) + y sqrt(-b))^3 for
some p, q. But Dickson goes on to say that in the same chapter of his
Algebra (Articles 195-196) Euler himself demonstrates that the assumption
is invalid in general! (Specifically, Euler shows that it fails for
a=2, x=4, b=-5, y=1.)
So, what did Euler believe he'd done? And what did his colleagues and
successors think he'd done?
Judging from what I've written above, you might suppose that it would
have been common knowledge that Euler's proof was incomplete. However,
Dickson tells us that in 1875 "T. Pepin criticized Euler's proofs, noting
that there may exist sets of formulas for x and y other than the set
deduced by Euler's assumption." (Dickson does not say whether Pepin
acknowledged that Euler himself had given a counterexample.)
So, how did Euler's proof come to be judged definitive when he himself
apparently did not deem it so?
Perhaps the passage in Euler's Algebra will help resolve the question.
Have any of you read it? Does Euler make a strong claim and then back off
from it later? (And have any of you ever looked into Pepin?)
Thanks,
Jim