[math-fun] Fermat's last theorem

Hi, Just a quick thought I had the other day, as I understand it Fermat's last theorem (now proved) basically says that for: a^p + b^p = c^p Then where all variables are integers there are no solutions for a, b and c where p>2. My thought was has anyone considered: a^p + b^p + c^p = d^p or indeed: a1^p + a2^p + a3^p + ..... an^p = b^p And is it possible that for the case of: a^p + b^p + c^p = d^p Then there is a solution for a,b,c,d for p=3 but not for p>3 and generally for: a1^p + a2^p + a3^p + ..... an^p = b^p there's a solution for a1..an and b if p=n but not for p>n ?