ajuda

[Luis Lopes]

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----- De: Filho Para: discussão de problemas 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