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Ai' vai a soluc,a~o do prof. Rousseau para o
problema
Prove que não existem inteiros positivos a,b e c tais
que:
a ^ 2 + b ^ 2 + c ^ 2 = a ^ 2 . b ^
2
[ ]'s
Lui's
Dear Luis:
The second problem was given on the USA Mathematical Olympiad in
1976. Suppose that there is a solution. Writing the equation as
(a^2 - 1)(b^2 - 1) = c^2 + 1, we see that if either a or b is odd then
the LHS is congruent to 0 (mod 4), but the RHS is congruent to either
1
or 2 (mod 4). Thus both a and b are even, and then c must be even as well.
Let k be the highest power of 2 that commonly divides a,b,c. Then
a = 2^k r, b = 2^k s, c = 2^k t where at least one of r,s,t is
odd and
r^2 + s^2 + t^2 = 4 r^2 s^2. But this is impossible since the
LHS is
congruent to 1, 2, or 3 (mod 4) while the RHS is congruent to 0 (mod
4).
I wasn't quite sure what the other question was abou.
Cecil
-----Mensagem Original-----
Enviada em: Domingo, 13 de Agosto de 2000
12:00
Assunto: ajuda
Prove que não existem inteiros positivos a,b e c tais
que:
a ^ 2 + b ^ 2 + c ^ 2 = a ^ 2 . b ^ 2
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