Any person, fond of mathematics, could not pass puzzle Fermat's Last Theorem. And I too. Its modern proof was published by Andrew Wiles in 1994, but the author acknowledged that such a mathematical apparatus, as he used  (the proof is based on properties of elliptic functions), Fermat could not yet be. That is, the original proof (if it was), has not yet been found. Apparently, Fermat caught some property of a numerical series, we are not clear. Thus, the theorem states that:

For any integer n> 2 (3,4,5, ...), the equation

aⁿ+bⁿ=cⁿ

 has no positive solutions a, b , c.

Ironically, most researchers immediately tries to refute the theory, trying to find "the right combination" aⁿ+bⁿ=cⁿ.

While I still at university, having tried the available computing power then I realized it was a dead end road.

Not so long ago I made a simple observation indicates intuitive faithfulness theorems. The fact that an infinite number of numerical 1,2,3,4,5,6 .... ∞ mark the degree of integers, the further from zero, the less likely they will occur. Take a degree of 5: 5²=25, 5³=125, 5⁴= 625, 5⁵=3125, … 5¹⁰= 9 765 625 ,…, 5²⁰=95 367 431 640 625…That is, from 95 trillion (!) Numbers will be only 5 (!!!) 20th degree of integer: 1²⁰, 2²⁰, 3²⁰, 4²⁰,5²⁰

That is to say that the probability of the degree of natural numbers on the numeric number is reduced. This probability P is equal to:

P(aⁿ)=a/aⁿ= 1/a^n-1

It is obvious that in the limit at infinity (as a-> ∞, n-> ∞), this probability tends to zero, P-> 0:

Thus for  aⁿ+bⁿ:

P(aⁿ+bⁿ)=(a+b)/( aⁿ+bⁿ), где a,b,n – integer ,

and, in limit,

Thus, if the formula to reflect the of Fermat aⁿ+bⁿ=cⁿ in the space of probability, it is true, the probability of the sum of powers of positive integers of the same degree as other natural number at infinity tends to zero.

Of course, this can not be strict proof of Fermat's Last Theorem, but it shows that the scientist was right. But perhaps that was the basis of evidence about which Fermat wrote in the margin, "Arithmetic" Diophantus, because at about the same time he corresponded with Blaise Pascal about the basics of probability theory.