The extract from the reference to the Editorial Board from V.F.Zaitsev, Guest Editor, Professor of the Department of Mathematical Analysis at A.I. Gertsen's RGPU, Doctor of Physics and Mathematics, who represented the article by L.E. Abramov:

 

Invitation to discussion

 

We strongly recommend you to read new article of Leonid Abramov, where entropy method and probability approach, developed by author in first article ("Theorems on infinite sequences with definite qualities", Problems of nonlinear Analysis in Engineering Systems, No.1(17), vol.9, 2003), is applied for the proof of original theorem. It is new theorem, that is correcting well-known Euler's hypothesis. At present two counter-examples, refuting Euler's hypothesis, are constructed. And it is very interesting problem: to find the strong conditions, under which Euler's hypothesis will be justified. Theorem, presented in this article, solves this problem. I haven't found any mistakes in the statements. The proof and principles are supported by the original approach worked out by the author and mainly based on the probability theory and statistics principles. Not being the expert in the probability theory, the author of this review can not completely guarantee the correctness of presented work. But the results, announced in this work, are of a priority and completely unique character.

 

V.F.Zaitsev,

Professor of the Department of Mathematical Analysis

A.I. Gertsen's RGPU, Doctor of Physics and Mathematics

 

About theorem correcting Euler's hypothesis

L.E.Abramov

Saint-Petersburg, Russia

ale@mail.spbnit.ru

 

There is known Euler's hypothesis [1], which states thatљ n-power of a natural number may be represented as sum љofљљ n- powers of some natural numbers, i. e.:

_{љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ} љinteger)љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (1)

Euler's hypothesis. Diophant equation (1) has no solution in natural numbers for any index of , if .

For a long time Euler's hypothesis, though looking like truthful, was neither proved nor refuted. But in 1967 the equality was derived [2] on a computer, this equality contradicts to this hypothesis:

144⁵ = 27⁵ + 84⁵ + 110⁵ + 133⁵

Another example refuting the Euler's hypothesis was found in 1988:

20615673⁴ = 2682440⁴ + 15365639⁴ + 187960⁴

Let's take and prove, using the Theorem 2 from the work [3], the next theorem correcting Euler's hypothesis.

Theorem. For any fixed value of љpower index the equation (1) has no solutions љin natural numbers љif

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (2)

except the case, when at least one value , љsatisfies the equality , where љare natural numbers.

In the latter case the equation (1) has no solutions in natural numbers, if

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (3)

When there are several values of , љsatisfying the equality , with various values of , it is necessary to substitute the least one of them into the expression (3).

Proof.љ At first let's prove the theorem for the case, when the inequality , љis fulfilled with arbitrary natural value . We'll consider a series of natural numbers. Suppose that features set љ[3] consists of two features љandљ љљthat are given to љelements of sequence by conditions and . Now we'll determine the conditions and љby the following :

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ ,

where љare natural numbers.

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ ,

whereљ k, qљ are natural numbers.

It is evident that simultaneous fulfilment of the conditions љand љis equivalent to that of equation (1) with , љand .

The properties 1 and 2 of the sequences, to what the theorem 2 from the work [3] is applied, are evidently fulfilled. The probability of the event , consisting in that љnatural numbers љhaving a feature are extracted from the first љelements of the sequence by aљ random sample with recovering the volumeљ љasymptotically for , is equal to:

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (4)

It may be shown that the probability of the event, consisting in that the extracted numbers љhave the quality љasymptotically for , is proportional to :

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (5)

With the relationships (4), (5) the conditions of the theorem 2 from the work [3] will be fulfilled:

if the inequality (2) is fulfilled.

Thus, if the inequality (2) is fulfilledљ and љ, then the theorem 2 from the work [3] denies the existence of љnumbers, which simultaneously have the features љand , in any final number of the first elements of a natural series, that is equivalent to the absence of the equation (1) solutions among the first љnumbers of a natural series. The transit to the infinite set of natural numbers is similar to that in the proof ofљ the Great Fermat's theorem [3].

Now we'll try to prove the theorem for a case, when at least one value of љsatisfies the equality , with arbitrary natural As before, we'll consider a series of natural numbers with set of features , which includes three of them: , љand љwith corresponding conditions , љand , where:

It is obvious that simultaneous fulfilment of the conditions љand љis equivalent to that of the equation (1) with , љand .

As for (5) it is fulfilled:

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (6)

whereљ љљis the probability of the eventљ љconsisting of that the numbersљ љextracted by a random sample of volume љhave the feature .

Using the equations (4) and (6) it may be shown that the conditions ofљ the theorem (2) from the work [3] for the features andљ љare fulfilled, if the inequality (3) is fulfilled, because in this case:

Since simultaneously with the feature љthe numbers extracted by a random sample must also have the feature , then the inequalities (2) and (3) must be also fulfilled simultaneously.

It is obvious that the inequality (3) corresponds to the solution of the system of these inequalities (2) and (3), with , and the inequality (2) is such in other cases. The conclusion follows that the solutions of the equationљ (1) are absent within a final number љof the natural series first numbers, when the inequality (3) is fulfilled, if though one value of , љsatisfies the equality .

Since a boundary value , below which the equation (1) solutions can't exist, decreases with a decreasing of љpower index, then in the case of several values of љsatisfying the equality љwith various values of љit is necessary to substitute the least of -pertaining values into the equation (3). The further theorem proof ( for the infinite set of natural numbers) is completely similar to that given in the work [3] for the proof of the Great Fermat's theorem.

The theorem is proven

It must be noted that in the first example above, which refutes Euler's hypothesis, term , in the right part of the equation (1), may be presented by. According to the inequality (3), the equation (1) with љand љhas no solutions in natural numbers for , but not for , as the inequality (2) corresponding to Euler's hypothesis requires.

In the second example with љand љeach term of the equality (1) may be represented by a power with its index . From the inequality (3) it follows that the solutions in natural numbers of the equation (1) are absent for љand љwith , but not with , as the inequality (2) requires.

Also note that inequality љfollows from the condition (3) for љand .

Therefore in all cases, when the condition (3) is applicable (and it is applicable to ), then the equation (1) for љhas not integer solutions for , and in all other cases- for .

Thus the Great Fermat's theorem follows from the corrected Euler's hypothesis.

 

 

References

 

1.            й.н.чЙОПЗТБДПЧ. пУОПЧЩ ФЕПТЙЙ ЮЙУЕМ. -н. - м., жЙЪНБФЗЙЪ, 1952.

2.            L.J.Lander, T.R.Parkin. A counter example to Euler's sum of power conjecture. Math. Comp., 21, 101-103, 1967.

3.            м.е.бВТБНПЧ. фЕПТЕНЩ П ВЕУЛПОЕЮОЩИ РПУМЕДПЧБФЕМШОПУФСИ У ПРТЕДЕМЕООЩНЙ УЧПКУФЧБНЙ. рТПВМЕНЩ ОЕМЙОЕКОПЗП БОБМЙЪБ Ч ЙОЦЕОЕТОЩИ УЙУФЕНБИ, ї1(17), ФПН 9, 2003.

 

Abramov Leonid Efimovich, Graduate of Tbilisi State University, Ph. D (physical and mathematical sciences), Senior-Researcher. Scientific interests are theory of probability and theoretical physics.