Diophantus claimed (see problem D-11 on page 165 of J. Sesiano's "Books IV
to VII of Diophantos'") "it is not possible to find a square number such that,
dividing it into two parts and increasing it by each of the parts, we obtain
in both cases a square". He also asserted, in problem VI-14, that the number
15 is not a sum of two (rational) squares.
(I don't actually have access to a copy of Sesiano; I am taking this information
from the 1998 edition of Victor Katz's "A History of Mathematics", page 180.)
Katz remarks that Diophantus gives a proof of neither of these negative
propositions. Has anyone made an attempt to reconstruct proofs of the
sort that Diophantus might have found?
Katz mentions (apropos of the former proposition) that there's an easy proof
by congruence arguments, but he does not make the claim that this is how
Diophantus proceeded in proving either of these propositions.
It is of course possible that Diophantus did not see the need for, and did
not discover, anything resembling what we would call a proof.
2) Congruence arguments
In connection with the preceding question, I am curious to know the history
of congruence arguments. We see parity arguments in ancient Greek mathematics,
and a mod 3 argument in The Book of Squares by Leonardo of Pisa. Then we see
(or infer) such arguments in the work of Fermat, and very many of them in the
work of Euler, culminating in Gauss' introduction of congruence notation.
I'd be curious to know if anyone has tried to follow this strand through time.
I'm especially curious about the "state-of-the-art" at around the time of
Fermat, because Descartes' dismissive treatment of Fermat's work on numbers
that can be written as sums of three rational squares suggests that congruence
arguments in Diophantine analysis were no longer anything novel, and could be
farmed out to clerical assistants.
Jim Propp
Department of Mathematics
University of Wisconsin