Prove that there are no solutions in positive integers to the equation x^4 + y^4=100
Modulo 4, every integer to the 4th power is 0 or 1 ==> since x^4 +_ y^4 = 100 = 0 (mod 4), it has to be x, y = 0 or 2 (mod 4) ==> x,y are even and thus x = 2m, y = 2n ==> x^4 + y^4 = 16(m^4 + n^4) = 100...but this, of course, is absurd (check powers of 2 in both sides).
Tonio