3.1. Short and simple proof of Fermat's theorem
            (The Last Fermat's Theorem)

Theorem
For natural numbers N greater than  2 there are no natural numbers  a, b, c   greater than zero satisfying the equation    c^N = b^N + a^N.

Proof
For the equation

in order to eliminate secondary solutions, we additionally assume that N 
is a prime number >2 and  a, b, c  are natural numbers, relatively prime.

The following relation applies for odd numbers N:

which means that  c^N must be divisible by  a+b.

By introducing two integer variables:  s=a+b  and   x=s−c we obtain:

which applies for each natural  N > 0 and maintains the required (per 3.2)
divisibility of  c^N   by s.

The expression in square brackets in (3.3) could represent an integer number,
but only in the case when  s and  x have a common divisor w.

This can be written as  s=w‧ v  and   x=w‧ u, where  v, u  are natural numbers.

By inserting these relations to (3.3) we can notice that left side of the equation  (3.3)  contains a multiplier w^N, which should be also presented on the right side of the equation.

We can see that an existence of required common divisor of  s  and  x   does not cause that expression in sqaure brackets in (3.3) is an integer number; if w would be the greatest common divisor then  v  and  u   would be relatively prime, then (after taking w^N-1 outside the brackets) the expression in square brackets cannot be an integer number.
Should we still relentlessly search for for common divisiors of  v  and  u   we will conlude that such divsor must be  x , so then equation (3.3) will be expressed as:

where it is clearly visible that expression in square brackets cannot be an integer number.

Thus, the natural, non-zero numbers s and x satisfying (3.3) do not exist.

We have shown above that the theorem is true for N > 2, being a prime number and for all compound exponents N > 2 containing divisors which are prime numbers different than 2.

In conjunction with the fact, that the theorem is true for N= 4
(as it is known from Fermat's time, and this proof for example is included in
Sierpinski's "Theory of Numbers" - Monografie Matematyczne PAN, Tom 19),
the theorem is true for all integers N > 2.

(End of proof)