Simple proof of Fermat’s last theorem
A^(n) + B^(n) ‡ C^(n) for all positive integers A,B,C and n, were n>2
A,B,C must be relatively prime for a non trivial solution to any such equation, if any integer solution exists. That requires that one and only one of those bases has at least one factor of 2.
For any two smaller objects (A & B) of order n to be equal in quantity to that of a larger n ordered object (C), the smaller objects when contained within the larger (on the longest line between 2 most distant vertices of the largest object and each of the smaller objects oriented in congruence with those opposite vertices of the largest object) must OVERLAP in a union, the n order quantity of that union (O^(n)) (this happens to be the minimum amount of union possible ) will equal the nth order quantity of the largest object that is disjoint from both of the smaller objects.
It becomes useful to express:
A=a+O
B=b+O
C=a+b+O O being the linear Overlap or C-(a+b)
When A^(n) shares a vertex and orientation with C^(n) and B^(n) is also located within C^(n) at the most distance vertex of C^(n) from that of A^(n) then it could be expressed that the overlap or union O^(n) would be required to equal the disjointed expressed by the binomial expansion of the(a + b)^(n) minus (a^(n)+b^(n).)
For n=2, O^(2)=2ab
For n=3, O^(3)= 3(a^(2)b) + 3a(b^(2)
For n=5, O^(5)=5(a4)b+10(a^(3)(b^2)+10(a^(2)(b^3)+5a(b^(4)
Because of the content of any such algebraic expansion of O^(n) (of particular interest were n is prime) O will be a even quantity and require either the factor a or b to be even (but not both).
That even integer would be the only source of factors of 2 for O^(n) and be required to contain at least n factors of 2. As a result O^(n) would contain 2n factors of 2 requiring that same even integer to also contain 2n factors of 2.
In short, A^(n) + B^(n) ‡ C^(n) under the conditions stated because the union of the two smaller objects when they are contained within the larger object can not have integer equivalence (the same number of factors of 2) to that which is disjoint