
theorem :: Problem 60
  not ex n being Nat st
    n is perfect_power & n+1 is perfect_power &
      n+2 is perfect_power & n+3 is perfect_power
  proof
    assume ex n being Nat st
    n is perfect_power & n+1 is perfect_power &
      n+2 is perfect_power & n+3 is perfect_power; then
    consider n being Nat such that
A1: n is perfect_power & n+1 is perfect_power &
      n+2 is perfect_power & n+3 is perfect_power;
    consider k being Nat such that
A3: n = 4*k or n = 4*k+1 or n = 4*k+2 or n = 4*k+3
      by NUMBER02:24;
    per cases by A3;
    suppose n = 4 * k;
      hence thesis by A1,Four2NotPerfect;
    end;
    suppose n = 4 * k + 1; then
      n + 1 = 4 * k + 2;
      hence thesis by A1,Four2NotPerfect;
    end;
    suppose n = 4 * k + 2;
      hence thesis by A1,Four2NotPerfect;
    end;
    suppose n = 4 * k + 3; then
      n + 3 = 4 * (k+1) + 2;
      hence thesis by A1,Four2NotPerfect;
    end;
  end;
