The FLT cannot be of the form

(8x+3)^m + (8y+5)^m = z^m  for even m

Since by Theorem 2, 2 is a once divisor of  (8x+3)^m + (8y+5)^m = z^m. This implies 2 divides z = 2k. Then z^m = (2^m)(k^m) ==> 2 is an m>1 divisor of the expression (8x+3)^m + (8y+5)^m contrary to theorem 2. Thus the FLT is proved for numbers of the form (8x+3) and  (8y+5) for all m>1.