Prove that if a²+b² is a multiple of ab+1, for positive integer a and b, then (a²+b²)/(ab+1) is a perfect square.
Proving that "if a^{2}+b^{2} is a multiple of ab+1, for positive integer a and b, then (a^{2}+b^{2})/(ab+1) is a perfect square" is a little beyond my current mathematical skill. But here is proof that when a = b^{3} (for positive integer a and b) it can be seen that (a^{2}+b^{2})/(ab+1) = b^{2.}
Substituting b^{3} for a, ((b^{3})^{2}+b^{2})/((b^{3})*b+1) = b^{2}
> (b^{6} + b^{2}) = b^{2}*(b^{4} + 1)
> (b^{6} + b^{2}) = b^{6} + b^{2}

Posted by Dej Mar
on 20070325 21:02:08 