Proof of Fermat Last Theorem based on successive presentations of pairs of odd numbers. Properties of equation x^a+y^a=z^b

A simpler proof of Fermat Last Theorem (FLT), formulated by Fermat in 1637, is suggested. The
initial equation x^n + y^n = z^n is considered not in natural, but in integer numbers. It is subdivided
into four equations based on parity of terms and their powers. All cases converge to one equation,
which is studied using presentation of pairs of odd integers with a successively increasing
presentation factor of 2^r. At each presentation level, the equation has no solution for a certain
subset of pairs of odd integers. Using introduced measure of such "no solution" subsets, we sum up
the corresponding measures across subsequent presentation levels, and prove that this sum
corresponds to all possible pairs of odd integers. Based on this result, we eventually prove that FLT
equation has no integer solution. The proposed approach also allowed to prove that equation x^n +
y^n = z^k has no integer solution, except for one particular combination of parameters, when
solution is uncertain. The proposed methods and ideas can be used for studying other problems in
number theory.