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Articles and Publication    Mathematics, calculation, statistic PROBABILISTIC INTERPRETATION OF FERMA’S THEOREM.


© K. A. Sytin


Institute of Mathematics of Siberian Branch of Russian Academy of Sciences "Maltsev's Meetings", Novosibirsk, Russia

Ferma’s theorem is represented in the following way:


Let’s divide both parts (1) into zn:


Consider the obtained equation as probability of some complex event equal to 1:


Replace probabilities of opposite events:

Replace the equation (2) by the equivalent:



from which it follows that:

It is seen that x=z, y=0, because x, y, are included into (1) symmetrically. So arguments and conclusions concerning y are similar, i.e. y=z, x=0. Thus, if the equation (1) is right , it is trivial (x=z, y=0 or y=z, x=0). And on the contrary, if the equation is trivial, it is correct. Therefore, in an untrivial case :

One should take into consideration that Ferma and Pascal were the pioneers of a new aria of mathematics – the theory of probability. They received the first evidence concerning this theory. It is quite natural to assume that “the most surprising” evidence which was not represented on the books could be probabilistic. In any case such an assumption should not be excluded.

The method chosen by us makes this theorem quite clear and gives the possibility to generalize it in any finite number of terms :

The analysis will be the same.

One can assume that Ferma’s theorem is essentially a probabilistic statement but it is formulated for beyond the theory of probabilities, namely, in the theory of numbers.

Naturally the question arises: “Can one consider the above mentioned derivation as the proof of the theorem ?”

In our opinion, there can be two answers – both positive and negative. We shall try to explain. From the point of view of arithmetical axiomatic the derivation given above are not the proof as we did not use the axioms of arithmetic but from the point of view of probabilities the proof is quite conclusive because the theorems of addition (2 and 2.1) and multiplication (3 and 3.1 ) of probabilities were used.

Thus, the difficulty in providing the theorem is due to the fact that it is transferred from one area of mathematics into another one. In this case the problem deals with revealing consistency or,. On the contrary, inconsistency of the systems of axioms of arithmetic and the theory of probabilities. So, the conclusion is the following: there appears a contradiction between the mentioned systems of axioms.

In fact, all our arguments can be repeated for n=2 and n=1 . The conclusions will be similar, i.e. the equation and x+y=z should be correct only in trivial cases. But in reality it is not so: and 3+4=7 - are counterexample to Ferma’s theorem which is considered in the probability area. That is why Ferma introduced the condition n ³ 3 .

One can assume that Ferma understood this contradiction. And it was that reason which made him search for a purely arithmetic proof of the hypothesis formulated by him (“the method of infinite descent”).

However, imagine the complete group of events the sum of which is (where p - probabilities) (4)

Then using the same approach one can come to the conclusion that the equation (4) should be correct only in a trivial case, i.e. some the others are equal to 0. (It should be noted that references to the theorems of addition and multiplication of probabilities in this case are the same as the reference to one of de Morgan’s rules).

From our point of view Ferma’s theorem is a unique event in the history of mathematics.

From the conducted analysis there appears a necessity of some additional and deep investigation of truth of both arithmetical and probabilistic axiomatics. In fact, Pythagorean theorem is true in arithmetical and geometric systems but in the probabilistic variant it can be true only in trivial cases. There arises a question: “What is preferable ?”. Testing the truth of Pythagorean theorem, as we imagine, can be accomplished in a physical experiment. But we don’t know if Pythagorean theorem was tested by such a method. At a macroscopic level the test cannot be considered quite exact because of the wave properties of light. One cannot determine quite exactly the distance between the peaks of the triangle by optical means. As we right know nobody conducted the precision physical experiments for checking Pythagorean theorem. Although some indirect estimates of the truth of Euclid geometry at a microscopic level are not excluded. The same can be said about cosmogonical scales as well.

Publishing date: October 6, 2003



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