﻿ Non-Archimedean Analysis on the Extended Hyperreal Line *R<sub>d</sub> and the Solution of Some Very Old Transcendence Conjectures over the Field Q

Vol.05 No.10(2015), Article ID:58868,41 pages
10.4236/apm.2015.510056

Non-Archimedean Analysis on the Extended Hyperreal Line and the Solution of Some Very Old Transcendence Conjectures over the Field Jaykov Foukzon

Center for Mathematical Sciences, Israel Institute of Technology, Haifa, Israel

Email: jaykovfoukzon@list.ru  Received 9 May 2015; accepted 15 August 2015; published 19 August 2015 ABSTRACT

In 1980 F. Wattenberg constructed the Dedekind completion of the Robinson non-archime- dean field and established basic algebraic properties of . In 1985 H. Gonshor estab- lished further fundamental properties of . In  important construction of summation of countable sequence of Wattenberg numbers was proposed and corresponding basic properties of such summation were considered. In this paper the important applications of the Dedekind completion in transcendental number theory were considered. Given any analytic function of one complex variable we investigate the arithmetic nature of the values of at transcendental points . Main results are: 1) the both numbers and are irrational; 2) number is transcendental. Nontrivial generalization of the Lindemann-Weierstrass theorem is obtained.

Keywords:

Non-Archimedean Analysis, Robinson Non-Archimedian Field, Dedekind Completion, Dedekind Hyperreals, Wattenberg Embeding, Gonshor Idempotent Theory, Gonshor Transfer 1. Introduction

In 1873 French mathematician, Charles Hermite, proved that is transcendental. Coming as it did 100 years after Euler had established the significance of , this meant that the issue of transcendence was one mathematicians could not afford to ignore. Within 10 years of Hermite’s breakthrough, his techniques had been extended by Lindemann and used to add to the list of known transcendental numbers. Mathematician then tried to prove that other numbers such as and are transcendental too, but these questions were too difficult and so no further examples emerged till today’s time. The transcendence of has been proved in 1929 by A. O. Gel’fond.

Conjecture 1. Whether the both numbers and are irrational.

Conjecture 2. Whether the numbers and are algebraically independent.

However, the same question with and has been answered:

Theorem. (Nesterenko, 1996  ) The numbers and are algebraically independent.

Throughout of 20-th century,a typical question: whether is a transcendental number for each algebraic number has been investigated and answered many authors .Modern result in the case of entire functions satisfying a linear differential equation provides the strongest results, related with Siegel’s E-functions   , ref  contains references to the subject before 1998, including Siegel and functions.

Theorem. (Siegel C. L.) Suppose that

(1.1)

Then is a transcendental number for each algebraic number

Let be an analytic function of one complex variable

Conjecture 3. Whether is an irrational number for given transcendental number

Conjecture 4. Whether is a transcendental number for given transcendental number

In this paper we investigate the arithmetic nature of the values of at transcendental points

Definition 1.1. Let be any real analytic function such that

(1.2)

We will call any function given by Equation (1.2) -analytic function and denoted by

Definition 1.2.   . A transcendental number is called #-transcendental number over the field, if there does not exist -analytic function such that i.e. for every -analytic function the inequality is satisfies.

Definition 1.3.   . A transcendental number is called w-transcendental number over the field, if is not #-transcendental number over the field, i.e. there exists -analytic function such that

Example. Number is transcendental but number is not -transcendental number over the field as

(1) function is a -analytic and

(2) i.e.

(1.3)

Main results are.

Theorem 1.1.   . Number is #-transcendental over the field.

From theorem 1.1 immediately follows.

Theorem 1.2. Number ee is transcendental.

Theorem 1.3.   . The both numbers and are irrational.

Theorem 1.4. For any number is #-transcendental over the field.

Theorem 1.5.   . The both numbers and are irrational.

Theorem 1.6.   . Let be a polynomials with coefficients in.

Assume that for any algebraic numbers over the field, form a complete set of the roots of such that

(1.4)

and. Assume that

(1.5)

Then

(1.6)

2. Preliminaries. Short Outline of Dedekind Hyperreals and Gonshor Idempotent Theory

Let be the set of real numbers and a nonstandard model of  . is not Dedekind complete.

For example, and are bounded subsets of which have no suprema or infima in

. Possible completion of the field can be constructed by Dedekind sections   . In  Wattenberg constructed the Dedekind completion of a nonstandard model of the real numbers and applied the construction to obtain certain kinds of special measures on the set of integers. Thus was established that the Dedekind com- pletion of the field is a structure of interest not for its own sake only and we establish further im- portant applications here. Important concept introduced by Gonshor  is that of the absorption number of an ele- ment which, roughly speaking, measures the degree to which the cancellation law fails for.

2.1. The Dedekind Hyperreals

Definition 2.1. Let be a nonstandard model of  and the power set of.

A Dedekind hyperreal is an ordered pair that satisfies the next conditions:

1. 2. 3.

4. 5.

Compare the Definition 2.1 with original Wattenberg definition  ,(see  def.II.1).

Designation 2.1. Let We designate in this paper

Designation 2.2. Let We designate in this paper

Remark 2.1. The monad of is the set: is denoted by.

Supremum of is denoted by. Supremum of is denoted by. Note that 

Let be a subset of bounded above. Then exists in  .

Example 2.1. 1), 2)

Remark 2.2. Unfortunately the set inherits some but by no means all of the algebraic structure on. For example, is not a group with respect to addition since if denotes the addition in then:

Thus is not even a ring but pseudo-ring only.

Definition 2.2. We define:

1. The additive identity (zero cut) often denoted by or simply 0 is

2. The multiplicative identity often denoted by or simply 1 is

Given two Dedekind hyperreal numbers and we define:

3. Addition of and often denoted by is

It is easy to see that for all

It is easy to see that is again a cut in and

Another fundamental property of cut addition is associativity:

This follows from the corresponding property of.

4. The opposite of, often denoted by or simply by, is

5. We say that the cut is positive if or negative if

The absolute value of, denoted, is, if and if

6. If then multiplication of and often denoted is

In general, if or, if or if or

7. The cut order enjoys on the standard additional properties of:

(i) transitivity:

(ii) trichotomy: eizer or but only one of the three

(iii) translation:

2.2. The Wattenberg Embeding into

Definition 2.3.  . Wattenberg hyperreal or #-hyperreal is a nonepty subset such that:

(i) For every and

(ii)

(iii) has no greatest element.

Definition 2.4.  . In paper  Wattenberg embed into by following way:

If the corresponding element, of is

(2.1)

Remark 2.3.  . In paper  Wattenberg pointed out that condition (iii) above is included only to avoid nonuniqueness. Without it would be represented by both and

Remark 2.4.  . However in paper  H. Gonshor pointed out that the definition (2.1) in Wattenberg paper  is technically incorrect. Note that Wattenberg  defines in general by

(2.2)

If i.e. has no mininum, then there is no any problem with definitions (2.1) and (2.2). However if for some i.e. then according to the latter definition (2.2)

(2.3)

whereas the definition of requires that:

(2.4)

Remark 2.5. Note that in the usual treatment of Dedekind cuts for the ordinary real numbers both of the latter sets are regarded as equivalent so that no serious problem arises  .

Remark 2.6. H. Gonshor  defines by

(2.5)

Definition 2.5. (Wattenberg embeding) We embed into of the following way: (i) if the corresponding element of is

(2.6)

and

(2.7)

or in the equivalent way,i.e. if the corresponding element of is

(2.8)

Thus if then where

(2.9)

Such embeding into Such embeding we will name Wattenberg embeding and to designate by.

Lemma 2.1.  .

(i) Addition is commutative and associative in.

(ii)

(iii)

Remark 2.7. Notice, here again something is lost going from to since does not imply since but

Lemma 2.2.  .

(i) a linear ordering on often denoted, which extends the usual ordering on.

(ii)

(iii)

(iv) is dense in. That is if in there is an then

(v) Suppose that is bounded above then exist in.

(vi) Suppose that is bounded below then exist in.

Remark 2.8. Note that in general case In particular the formula for

given in  on the top of page 229 is not quite correct  , see Example 2.2. However by Lemma 2.2 (vi) this is no problem.

Example 2.2.  . The formula says

Let be the set where runs through the set of all positive numbers in, then However

Lemma 2.3.  .

(i) If then

(ii)

(iii)

(iv)

(v)

(vi)

Proof. (v) By (iv):

(1) Suppose now this means

(2) and therefore

(3)

(4) Note that: (since and imply but this is a contradiction)

(5) Thus and therefore

(6) By similar reasoning one obtains:

(7) Note that: and therefore

Lemma 2.4. (i)

(ii)

Proof. (i) For the statement is clear. Suppose now without loss of generality By Lemma 2.3. (iv):

(1) Suppose and therefore but this means

(2) and therefore

(3)

(4) Note that: (since and imply but this is a contradiction)

(5) Thus and therefore

(6) By similar reasoning one obtains:

(7) Note that: and therefore

(ii) Immediately follows from (i) by Lemma 2.3.

Definition 2.6. Suppose. The absolute value of written is defined as follows:

Definition 2.7. Suppose The product, is defined as follows: Case (1):

(2.10)

Case (2)

Case (3)

(2.11)

Lemma 2.5.  . (i)

(ii) Multiplication is associative and commutative:

(2.12)

(iii) where

(iv)

(v) (2.13)

(vi) (2.14)

Lemma 2.6. Suppose and Then

(2.15)

Proof. We choose now:

(1) such that:

(2) Note that

Then from (2) by Lemma 2.4. (ii) one obtains

(3) Therefore

(4)

(5) Then from (4) by Lemma 2.5. (v) one obtains

(6)

Then from (6) by Lemma 2.4. (ii) one obtains

(7)

Definition 2.8. Suppose then is defined as follows:

(i)

(ii)

Lemma 2.7.  .

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Lemma 2.8.  . Suppose that Then

Theorem 2.1. Suppose that is a non-empty subset of which bounded from above, i.e. exist and suppose that Then

(2.16)

Proof. Let Then is the smallest number such that, for any Let Since for any Hence is bounded above by Hence has a supremum Now we have to prove that Since is an upper bound for and is the smallest upper bound for, Now we repeat the argument above with the roles of and reversed. We know that is the smallest number such that, for

any Since it follows that for any But Hence is an upper bound for. But is a supremum for. Hence and We have shown that and also that Thus

2.3. Absorption Numbers in

One of standard ways of defining the completion of involves restricting oneself to subsets, which have the following property . It is well known that in this case we obtain a field. In fact the proof is essentially the same as the one used in the case of ordinary Dedekind cuts in the development of the standard real numbers, , of course, does not have the above property because no infinitesimal works.This suggests the introduction of the concept of absorption part of a number for an element of which, roughly speaking, measures how much departs from having the above property  .

Definition 2.9.  . Suppose then

(2.17)

Example 2.5.

(i)

(ii)

(iii)

(iv)

(v)

Lemma 2.9.  .

(i) and

(ii) and

Remark 2.9. By Lemma 2.7 may be regarded as an element of by adding on all negative elements of to. Of course if the condition in the definition of is deleted we automatically get all the negative elements to be in since The reason for our

definition is that the real interest lies in the non-negative numbers. A technicality occurs if. We then identify with 0. [becomes which by our early convention is not in].

Remark 2.10. By Lemma 2.7(ii), is additive idempotent.

Lemma 2.10.  .

(i) is the maximum element such that

(ii) for

(iii) If is positive and idempotent then

Lemma 2.11.  . Let satsify Then the following are equivalent. In what follows assume

(i) is idempotent,

(ii)

(iii)

(iv)

(v) for all finite

Theorem 2.2.  .

Theorem 2.3.  .

Theorem 2.4.  .

(i)

(ii)

Theorem 2.5.  . Suppose then

(i)

(ii)

Theorem 2.6.  . Assume If absorbs then absorbs.

Theorem 2.7.  . Let Then the following are equivalent

(i) is an idempotent,

(ii)

(iii)

(iv) Let and be two positive idempotents such that Then

2.4. Gonshor Types of a with Given ab.p.(a).

Among elements of such that one can distinguish two many different types following  .

Definition 2.10.  . Assume

(i) has type 1 if

(ii) has type 2 if i.e. has type 2 iff does not have type 1.

(iii) has type 1A if

(iv) has type 2A if

2.5. Robinson Part of Absorption Number

Theorem 2.8.  . Suppose Then there is a unique standard called Wattenberg stan- dard part of and denoted by such that:

(i)

(ii) implies

(iii) The map is continuous.

(iv)

(v)

(vi)

(vii) if

Theorem 2.9.  .

(i) has type 1 iff has type 1A,

(ii) cannot have type 1 and type 1A simultaneously.

(iii) Suppose Then has type 1 iff has the form for some.

(iv) Suppose. has type 1A iff has the form for some.

(v) If then has type 1 iff has type 1.

(vi) If then has type 2 iff either or has type 2.

Proof (iii) Let Then. Since (we chose such that and write as).

It is clear that works to show that has type 1.

Conversely, suppose has type 1 and choose such that: Then we claim that:

By definition of certainly. On the other hand by choice of, every element of has the form with.

Choose such that then.

Hence Therefore

Examples.

(i) has type 1 and therefore has type 1A. Note that also has type 2. (ii) Suppose Then has type 1 and therefore has type 1A.

(ii) Suppose i.e. has type 1 and therefore by Theorem 2.9 has the form for some unique Then, we define unique Robinson part of absor- ption number by formula

(2.18)

(iii) Suppose i.e. has type 1A and therefore by Theorem 2.9 has the form for some unique Then we define unique. Robinson part of absorption number by formula

(2.19)

(iv) Suppose and has type 1A, i.e. has the form for some Then, we define Robinson part of absorption number by formula

(2.20)

(v) Suppose and has type 1A, i.e. has the form for some Then, we define Robinson part of absorption number by formula

(2.21)

Remark 2.11. Note that in general case, i.e. if Robinson part of absorption number is not unique.

Remark 2.12. Suppose and has type 1 or type 1A. Then by definitions above one obtains the representation

2.6. The Pseudo-Ring of Wattenberg Hyperintegers

Lemma 2.12.  . Suppose that Then the following two conditions on are equivalent:

(i)

(ii)

Definition 2.11.  . If satisfies the conditions mentioned above is said to be the Wattenberg hyperinteger. The set of all Wattenberg hyperintegers is denoted by

Lemma 2.13.  . Suppose Then

(i)

(ii)

(iii)

The set of all positive Wattenberg hyperintegers is called the Wattenberg hypernaturals and is denoted by

Definition 2.12. Suppose that (i) (ii) and (iii)

If and satisfies these conditions then we say that is divisible by and we denote this by

Definition 2.13. Suppose that (i) and (ii) there exists such that

(1) or

(2)

If satisfies the conditions mentioned above then we say that is divisible by and we denote this by.

Theorem 2.10. (i) Let be a prime hypernaturals such that (i). Let be a Wattenberg hypernatural such that (i). Then

(ii) has type 1 iff has type 1A,

(iii) cannot have type 1 and type 1A simultaneously.

(iv) Suppose Then has type 1 iff has the form for some

(v) Suppose has type 1A iff has the form for some

(vi) Suppose If then has type 1 iff has type 1.

(vii) Suppose If then has type 2 iff either or has type 2.

Proof. (i) Immediately follows from definitions (2.12)-(2.13).

(iv) Let Then. Since (we chose such that and write a as).

It is clear that a works to show that has type 1.

Conversely, suppose has type and choose such that: Then we claim that:

By definition of certainly. On the other hand by choice of a, every element of has the form with.

Choose such that then.

Hence Therefore

2.7. The Integer Part Int.p(a) of Wattenberg Hyperreals

Definition 2.14. Suppose Then, we define by formula

Obviously there are two possibilities:

1. A set has no greatest element. In this case valid only the

Property I:

Since implies such that But then which implies contradicting

2. A set has a greatest element, In this case valid the

Property II: and obviously

Definition 2.15. Suppose Then, we define by formula

Note that obviously:

2.8. External Sum of the Countable Infinite Series in

This subsection contains key definitions and properties of summ of countable sequence of Wattenberg hyperreals.

Definition 2. 16.  . Let be a countable sequence such that

(i) or (ii) or

(iii)

Then external sum (#-sum) of the corresponding countable sequence is defined by

(2.22)

Theorem 2.11. (i) Let be a countable sequence such that and Then.

(ii) Let be a countable sequence such that and Then.

(iii) Let be a countable sequence such that and infinite series absolutely converges to in Then

(2.23)

(iv) Let be a countable sequence such that and infinite series absolutely converges to in Then

(2.24)

(v) Let be a countable sequence such that

(1) and

(2)

Then

(2.25)

Proof. (i) Let and Then obviously:

Thus there exists such that (1)

(1)

Therefore from (1) by Robinson transfer one obtains (2)

(2)

Using now Wattenberg embedding from (2) we obtain (3)

(3)

From (3) one obtains (4)

(4)

Note that obviously

(5)

From (4) and (5) one obtains (6)

(6)

Thus (i) immediately from (6) and from definition of the idempotent.

Proof.(ii) Immediately from (i) by Lemma 2.3 (v).

Proof.(iii) Let. Then obviously: and. Thus there exists such that (1)

(1)

Therefore from (1) by Robinson transfer one obtains (2)

(2)

Using now Wattenberg embedding from (2) we obtain (3)

(3)

From (3) one obtains (4)

(4)

From (4) by Definition 2.16 (i) one obtains

(5)

Note that obviously

(6)

From (5)-(6) follows (7)

(7)

Thus Equation (2.23) immediately from (7) and from definition of the idempotent.

Proof.(iv) Immediately from (iii) by Lemma 2.3 (v).

Proof.(v) From Definition 2.16.(iii) and Equation (2.23)-Equation (2.24) by Theorem 2.7.(iii) one obtains

Theorem 2.12. Let be a countable sequence such that and infinite series absolutely converges in. Let be external sum of the corresponding countable sequence. Let be a countable sequence where is any rearrangement of terms of the sequence. Then external sum of the corresponding countable sequence has the same value s as external sum of the countable sequence, i.e.

Theorem 2.13. (i) Let be a countable sequence such that (1) (2) infinite series absolutely converges to in and let be external sum of the corresponding sequence. Then for any the equality is satisfied

(2.26)

(ii) Let be a countable sequence such that (1) (2) infinite series absolutely converges to in and let be external sum of the corresponding sequence. Then for any the equality is satisfied:

(2.27)

(iii) Let be a countable sequence such that

(1)

(2) infinite series absolutely converges to in,

(3) infinite series absolutely converges to in.

Then the equality is satisfied:

(2.28)

Proof. (i) From Definition 2.16. (i) by Theorem 2.1, Theorem 2.11. (i) and Lemma (2.4) (ii) one obtains

(ii) Straightforward from Definition 2.16. (i) and Theorem 2.1, Theorem 2.11. (ii) and Lemma (2.4) (ii) one obtains

(iii) By Theorem 2.11. (iii) and Lemma (2.4). (ii) one obtains

But other side from (i) and (ii) follows

Definition 2.17. Let be a countable sequence such that infinite series absolutely converges in to We assume now that:

(i) there exists such that or

(ii) there exists such that or

(iii) there exists infinite sequence such that

(a) and infinite series absolutely converges in to and

(b) there exists infinite sequence such that and infinite series absolutely converges in to.

Then: (i) external upper sum (#-upper sum) of the corresponding countable sequence is defined by

(2.29)

(ii) external lower sum (#-lower sum) of the corresponding countable sequence is defined by

(2.30)

Theorem 2.14. (1) Let be a countable sequence such that infinite series absolutely converges in to We assume now that:

(i) there exists such that or

(ii) there exists such that or

(iii) there exists infinite sequence such that

(a) and infinite series absolutely converges in to and

(b) there exists infinite sequence such that and infinite series absolutely converges in to.

Then

(2.31)

and

(2.32)

Proof. (i), (ii), (iii) straightforward from definitions.

Theorem 2.15. (1) Let be a countable sequence such that infinite series absolutely converges in to We assume now that:

(i) there exists such that or

(ii) there exists such that or

(iii) there exists infinite sequence such that

(a) and infinite series absolutely converges in to and

(b) there exists infinite sequence such that and infinite series absolutely converges in to.

Then for any the equalities is satisfied

(2.33)

and

(2.34)

Proof. Copy the proof of the Theorem 2.13.

Theorem 2.16. (1) Let be a countable sequence such that infinite series absolutely converges in to We assume now that:

(i) there exists such that or

(ii) there exists such that or

(iii) there exists infinite sequence such that

(a) and infinite series absolutely converges in to and

(b) there exists infinite sequence such that and infinite series absolutely converges in to

Then for any the equalities is satisfied

(2.35)

and

(2.36)

Proof. (1) From Equation (2.31) we obtain

(2.37)

From Equation (2.37) by Theorem 2.1 we obtain directly

(2.38)

(2) From Equation (2.32) we obtain

(2.39)

From Equation (2.39) by Theorem 2.1 we obtain directly

(2.40)

Remark 2.13. Note that we have proved Equation (2.35) and Equation (2.36) without any reference to the Lemma 2.4.

Definition 2.18. (i) Let be a countable sequence such that

(2.41)

Then external countable upper sum (#-sum) of the countable sequence is defined by

(2.42)

In particular if where the external countable upper sum (#-sum) of the countable sequence is defined by

(2.43)

(ii) Let be a countable sequence such that

(2.44)

Then external countable lower sum (#-sum) of the countable sequence is defined by

(2.45)

In particular if where the external countable lower sum (#-sum) of the countable sequence is defined by

(2.46)

Theorem 2.17. (i) Let be a countable sequence such that valid the property (2.41). Then for any the equality is satisfied

(2.47)

(ii) Let be a countable sequence such that valid the property (2.44).

Then for any the equality is satisfied

(2.48)

Proof. Immediately from Definition 2.18 by Theorem 2.1.

Definition 2.19. Let be a countable sequence such that infinite series absolutely converges in. Then external countable complex sum (#-sum) of the corresponding countable sequence is defined by

(2.49)

correspondingly.

Note that any properties of this sum immediately follow from the properties of the real external sum.

Definition 2.20. (i) We define now Wattenberg complex plane by with. Thus for any we obtain, where, (ii) for any such that we define by.

Theorem 2.18. Let be a countable sequence such that infinite series absolutely converges in to and. Then

(i)

(ii)

2.9. Gonshor Transfer

Definition 2.21.  . Let.

Note that satisfies the usual axioms for a closure operator,i.e. if (i) and

(ii) S has no maximum, then

Let f be a continuous strictly increasing function in each variable from a subset of into. Specifically, we want the domain to be the cartesian product where for some By Robin- son transfer f extends to a function from the corresponding subset of into which is also strictly increasing in each variable and continuous in the Q topology (i.e. and range over arbitrary positive elements in). We now extend to

(2.50)

Definition 2.22.  . Let then

(2.51)

Theorem 2.19.  . If f and g are functions of one variable then

(2.52)

Theorem 2.20.  . Let f be a function of two variables. Then for any and

(2.53)

Theorem 2.21.  . Let f and g be any two terms obtained by compositions of strictly increasing continuous functions possibly containing parameters in. Then any relation or valid in extends to i.e.

(2.54)

Remark 2.14. For any function we often write for short instead of.

Theorem 2.22.  . (1) For any

(2.55)

For any

(2.56)

(2) For any

(2.57)

(3) For any

(2.58)

(4) For any

(2.59)

Note that we must always beware of the restriction in the domain when it comes to multiplication.

Theorem 2.23.  . The map maps the set of additive idempotents onto the set of all multiplicative idempotents other than 0.

3. The Proof of the #-Transcendene of the Numbers

In this section we will prove the #-transcendence of the numbers Key idea of this proof reduction of the statement of is #-transcendental number to equivalent statement in by using pseudoring of Wattenberg hyperreals  and Gonshor idempotent theory  . We obtain this reduction by three steps, see Subsections 3.2.1 - 3.2.3.

3.1. The Basic Definitions of the Shidlovsky Quantities

In this section we remind the basic definitions of the Shidlovsky quantities  . Let and be the Shidlovsky quantities:

(3.1)

(3.2)

(3.3)

where this is any prime number. Using Equations (3.1)-(3.3.) by simple calculation one obtains:

(3.4)

and consequently

(3.5)

Lemma 3.1.  . Let p be a prime number. Then

Proof. ( , p. 128) By simple calculation one obtains the equality

(3.6)

where p is a prime. By using equality where from Equations (3.1) and (3.6) one obtains

(3.7)

Thus

(3.8)

Lemma 3.2.  . Let p be a prime number. Then .

Proof. ( , p. 128) By subsitution from Equation (3.3) one obtains

(3.9)

By using equality

(3.10)

and by subsitution Equation (3.10) into RHS of the Equation (3.9) one obtains

(3.11)

Lemma 3.3.  . (i) There exists sequences and such that

(3.12)

where sequences and does not depend on number p. (ii) For any if.

Proof. ( , p. 129) Obviously there exists sequences and such that and does not depend on number p

(3.13)

and

(3.14)

Substitution inequalities (3.13)-(3.14) into RHS of the Equation (3.3) by simple calculation gives

(3.15)

Statement (i) follows from (3.15). Statement (ii) immediately follows from a statement (ii).

Lemma 3.4.  . For any and for any such that there exists such that

(3.16)

Proof. From Equation (3.5) one obtains

(3.17)

From Equation (3.17) by using Lemma 3.3. (ii) one obtains (3.17).

Remark 3.1. We remind now the proof of the transcendence of following Shidlovsky proof is given in his book  .

Theorem 3.1. The number is transcendental.

Proof. ( , pp. 126-129) Suppose now that is an algebraic number; then it satisfies some relation of the form

(3.18)

where integers and where Having substituted RHS of the Equation (3.5) into Equation (3.18) one obtains

(3.19)

From Equation (3.19) one obtains

(3.20)

We rewrite the Equation (3.20) for short in the form

(3.21)

We choose now the integers such that:

(3.22)

and. Note that Thus one obtains

(3.23)

and therefore

(3.24)

By using Lemma 3.4 for any such that we can choose a prime number such that:

(3.25)

From (3.25) and Equation (3.21) we obtain

(3.26)

From (3.26) and Equation (3.24) one obtains the contradiction.This contradiction finalized the proof.

3.2. The Proof of the #-Transcendene of the Numbers. We Will Divide the Proof into Four Parts

3.2.1. Part I. The Robinson Transfer of the Shidlovsky Quantities

In this subsection we will replace using Robinson transfer the Shidlovsky quantities by corresponding nonstandard quantities The properties of the nonstandard quantities one obtains directly from the pro-

perties of the standard quantities using Robinson transfer principle   .

1. Using Robinson transfer principle   from Equation (3.8) one obtains directly

(3.27)

From Equation (3.11) using Robinson transfer principle one obtains:

(3.28)

Using Robinson transfer principle from inequality (3.15) one obtains:

(3.29)

Using Robinson transfer principle, from Equation (3.5) one obtains:

(3.30)

Lemma 3.5. Let, then for any and for any there exists such that

(3.31)

Proof. From Equation (3.30) we obtain:

(3.32)

From Equation (3.32) and (3.29) we obtain (3.31).

3.2.2. Part II. The Wattenberg Imbedding into

In this subsection we will replace by using Wattenberg imbedding  and Gonshor transfer the nonstandard quantities and the nonstandard Shidlovsky quantities by correspond- ing Wattenberg quantities The properties of the Wattenberg quantities one obtains directly from the properties of the co- rresponding nonstandard quantities using Gonshor transfer principle   .

1. By using Wattenberg imbedding from Equation (3.30) one obtains

(3.33)

2. By using Wattenberg imbedding and Gonshor transfer (see Subsection 2.9 Theorem 2.19) from Equation (3.27) one obtains

(3.34)

3. By using Wattenberg imbedding from Equation (3.28) one obtains

(3.35)

Lemma 3.6. Let then for any and for any there exists such that

(3.36)

Proof. Inequality (3.36) immediately follows from inequality (3.31) by using Wattenberg imbedding and Gonshor transfer.

3.2.3. Part III. Reduction of the Statement of e Is #-Transcendental Number to Equivalent Statement in Using Gonshor Idempotent Theory

To prove that is #-transcendental number we must show that e is not w-transcendental, i.e., there does not exist real -analytic function with rational coefficients such that

(3.37)

Suppose that e is w-transcendental, i.e., there exists an -analytic function with rational coefficients:

(3.38)

such that the equality is satisfied:

(3.39)

In this subsection we obtain an reduction of the equality given by Equation (3.39) to equivalent equality given by Equation (3). The main tool of such reduction that external countable sum defined in Subsection 2.8.

Lemma 3.7. Let and be the sum correspondingly

(3.40)

Then

Proof. Suppose there exists k such that Then from Equation (3.39) follows There- fore by Theorem 3.1 one obtains the contradiction.

Remark 3.2. Note that from Equation (3.39) follows that in generel case there is a sequence such that

(3.41)

or there is a sequence such that

(3.42)

or both sequences and with a property that is specified above exist.

Remark 3.3. We assume now for short but without loss of generelity that (3.41) is satisfied. Then from (3.41) by using Definition 2.17 and Theorem 2.14 (see Subsection 2.8) one obtains the equality 

(3.43)

Remark 3.4. Let and be the upper external sum defined by

(3.44)

Note that from Equation (3.43)-Equation (3.44) follows that

(3.45)

Remark 3.5. Assume that and. In this subsection we will write for a short iff absorbs, i.e.

Lemma 3.8.

Proof. Suppose there exists such that Then from Equation(3.45) one obtains

(3.46)

From Equation (3.46) by Theorem 2.11 follows that and therefore by Lemma 3.7 one obtains the contradiction.

Theorem 3.2.  The equality (3.43) is inconsistent.

Proof. Let us consider hypernatural number defined by countable sequence

(3.47)

From Equation (3.43) and Equation (3.47) one obtains

(3.48)

Remark 3.6. Note that from inequality (3.27) by Wattenberg transfer one obtains

(3.49)

Substitution Equation (3.30) into Equation (3.48) gives

(3.50)

Multiplying Equation (3.50) by Wattenberg hyperinteger by Theorem 2.13 (see subsec- tion 2.8) one obtains

(3.51)

By using inequality (3.49) for a given we will choose infinite prime integer such that:

(3.52)

Now using the inequality (3.49) we are free to choose a prime hyperinteger and in the Equation (3.51) for a given such that:

(3.53)

Hence from Equation (3.52) and Equation (3.53) we obtain

(3.54)

Therefore from Equations (3.51) and (3.54) by using definition (2.15) of the function given by Equation (2.20)-Equation (2.21) and corresponding basic property I (see Subsection 2.7) of the function we obtain

(3.55)

From Equation (3.55) using basic property I of the function finally we obtain the main equality

(3.56)

We will choose now infinite prime integer in Equation (3.56) such that

(3.57)

Hence from Equation (3.34) follows

(3.58)

Note that Using (3.57) and (3.58) one obtains:

(3.59)

Using Equation(3.35) one obtains

(3.60)

3.2.4. Part IV. The Proof of the Inconsistency of the Main Equality (3.56)

In this subsection we wil prove that main equality (3.56) is inconsistent. This prooff based on the Theorem 2.10 (v), see Subsection 2.6.

Lemma 3.9. The equality (3.56) under conditions (3.59)-(3.60) is inconsistent.

Proof. (I) Let us rewrite Equation (3.56) in the short form

(3.61)

where

(3.62)

From (3.59)-(3.60) follows that

(3.63)

Remark 3.7. Note that Otherwise we obtain that But the other hand from Equation (3.61) follows that But this is a contradic- tion. This contradiction completed the proof of the statement (I).

(II) Let and be the external sum correspondingly

(3.64)

Note that from Equation (3.61) and Equation (3.64) follows that

(3.65)

Lemma 3.10. Under conditions (3.59)-(3.60)

(3.66)

and

(3.67)

Proof. First note that under conditions (3.59)-(3.60) one obtains

(3.68)

Suppose that there exists an such that Then from Equation (3.65) one obtains

(3.69)

From Equation (3.69) by Theorem 2.17 one obtains

(3.70)

Thus

(3.71)

From Equation (3.71) by Theorem 2.11 follows that and therefore by Lemma 3.7 one obtains the contradiction. This contradiction finalized the proof of the Lemma 3.10.

Part (III)

Remark 3.8. (i) Note that from Equation (3.62) by Theorem 2.10 (v) follws that has the form

(3.72)

where

(3.73)

(ii) Substitution by Equation (3.72) into Equation (3.61) gives

(3.74)

Remark 3.9. Note that from (3.74) by definitions follows that

(3.75)

Remark 3.10. Note that from (3.73) by construction of the Wattenberg integer obviously follows that there exists some such that

(3.76)

Therefore

(3.77)

Note that under conditions (3.59)-(3.60) and (3.73) obviously one obtains

(3.78)

From Equation(3.74) follows that

(3.79)

Therefore

(3.80)

From (3.78) follows that

(3.81)

Note that by Theorem 2.8 (see Subsection 2.5) and Formula (3.44) one otains

(3.82)

From Equation (3.81)-Equation (3.82) follows that

(3.83)

Thus

(3.84)

and therefore

(3.85)

But this is a contradiction. This contradiction completed the proof of the Lemma 3.9.

4. Generalized Shidlovsky Quantities

In this section we remind the basic definitions of the Shidlovsky quantities, see  pp. 132-134.

Theorem 4.1.  Let be a polynomials with coefficients in. Assume that for any algebraic numbers over the field form a complete set of the roots of such that

(4.1)

and Then:

(4.2)

Let be a polynomial such that

(4.3)

Let and be the quantities  :

(4.4)

where in (4.4) we integrate in complex plane along line see Picture 1 .

(4.5)

where and where in (4.5) we integrate in complex plane along line with initial point and which are parallel to real axis of the complex plane, see Picture 1 .

(4.6)

where and where in (4.6) we integrate in complex plane along contour see Picture 1 .

From Equation (4.3) one obtains

(4.7)

where Now from Equation (4.4) and Equation (4.7) using formula

Picture 1. Contour in complex plane.

one obtains

(4.8)

where We choose now a prime p such that Then from Equation (4.8) follows that

(4.9)

From Equation (4.3) and Equation (4.5) one obtains

(4.10)

where By change of the variable integration in RHS of the Equation (4.10) we obtain

(4.11)

where Let us rewrite now Equation (4.11) in the following form

(4.12)

Let be a ring of the all algebraic integers. Note that 

(4.13)

Let us rewrite now Equation (4.12) in the following form

(4.14)

where From Equation (4.14) one obtains

(4.15)

The polinomial is a symmetric polynomial on any system of variables where

(4.16)

It well known that  and therefore

(4.17)

From Equation (4.15) and Equation (4.17) one obtains

(4.18)

Therefore

(4.19)

Let be a circle wth the centre at point. We assume now that. We will designate now

(4.20)

From Equation (4.6) and Equation (4.20) one obtains

(4.21)

where Note that

(4.22)

From (4.22) follows that for any there exists a prime number p such that

(4.23)

where From Equation (4.4)-Equation (4.6) follows

(4.24)

where Assume now that

(4.25)

Having substituted RHS of the Equation (4.24) into Equation (4.25) one obtains

(4.26)

From Equation (4.26) by using Equation (4.19) one obtains

(4.27)

We choose now a prime such that and Note that and therefore from Equation (4.19) and Equation (4.27) one obtains the contradiction. This contradiction com- pleted the proof.

5. Generalized Lindemann-Weierstrass Theorem

Theorem 5.1.  Let be a polynomials with coefficients in. Assume that for any algebraic numbers over the field form a complete set of the roots of such that

(5.1)

and. We assume now that

(5.2)

Then

(5.3)

We will divide the proof into three parts.

Part I. The Robinson transfer

Let be a nonstandard polynomial such that

(5.4)

Let, and be the quantities:

(5.5)

where in (5.5) we integrate in nonstandard complex plane along line see Picture 1 .

(5.6)

where and where in (5.6) we integrate in nonstandard complex plane along line with initial point and which are parallel to real axis of the complex plane, see Picture 1 .

(5.7)

where and where in (5.7) we integrate in nonstandard complex plane along contour see Picture 1 .

1. Using Robinson transfer principle  - from Equation (5.5) and Equation (4.8) one obtains directly

(5.8)

where We choose now infinite prime such that

(5.9)

2. Using Robinson transfer principle from Equation (5.6) and Equation (4.19) one obtains directly

(5.10)

and therefore

(5.11)

3. Using Robinson transfer principle from Equation (5.7) and Equation (4.21) one obtains directly

(5.12)

where Note that there exists

(5.13)

4. From (5.13) follows that for any there exists an infinite prime such that

(5.14)

where

5. From Equation (5.5)-Equation (5.7) we obtain

(5.15)

where

Part II. The Wattenberg imbedding into

1. By using Wattenberg imbedding and Gonshor transfer (see Subsection 2.8 Theorem 2.17) from Equation (5.8) one obtains

(5.16)

where We choose now an infinite prime such that

(5.17)

2. By using Wattenberg imbedding and Gonshor transfer from Equation (5.10) one obtains directly

(5.18)

and therefore

(5.19)

3. By using Wattenberg imbedding and Gonshor transfer from Equation (5.14) one obtains directly

(5.20)

4. By using Wattenberg imbedding and Gonshor transfer from Equation (5.15) one obtains directly

(5.21)

where

Part III. Main equality

Remark 5.1. Note that in this subsection we often write for a short instead For example we write

Assumption 5.1. Let be a polynomials with coefficients in. Assume that for any algebraic numbers over the field: form a complete set of the roots of such that

(5.22)

Note that from Assumption 5.1 follows that algebraic numbers over the field: for any form a complete set of the roots of

(5.23)

Assumption 5.2. We assume now that there exists a sequence

(5.24)

and rational number

(5.25)

such that

(5.26)

and

(5.27)

Assumption 5.3. We assume now for a short but without loss of generality that the all numbers are real.

In this subsection we obtain an reduction of the equality given by Equation (5.27) in to some equivalent equality given by Equation (3) in. The main tool of such reduction that external countable sum defined in Subsection 2.8.

Lemma 5.1. Let and be the sum correspondingly

(5.28)

Then

Proof. Suppose there exists r such that Then from Equation (5.27) follows There- fore by Theorem 4.1 one obtains the contradiction.

Remark 5.2. Note that from Equation (5.27) follows that in generel case there is a sequence such that

(5.29)

or there is a sequence such that

(5.30)

or both sequences and with a property that is specified above exist.

Remark 5.3. We assume now for short but without loss of generelity that (5.29) is satisfied. Then from (5.29) by using Definition 2.17 and Theorem 2.14 (see Subsection 2.8) one obtains the equality 

(5.31)

Remark 5.4. Let and be the upper external sum defined by

(5.32)

Note that from Equation (5.31)-Equation (5.32) follows that

(5.33)

Remark 5.5. Assume that and In this subsection we will write for a short iff absorbs, i.e.

Lemma 5.2.

Proof. Suppose there exists such that Then from Equation (5.33) one obtains

(5.34)

From Equation (5.34) by Theorem 2.11 follows that and therefore by Lemma 5.1 one obtains the contradiction.

Theorem 5.2.  The equality (5.31) is inconsistent.

Proof. Let us considered hypernatural number defined by countable sequence

(5.35)

From Equation (5.31) and Equation (5.35) one obtains

(5.36)

where

(5.37)

Remark 5.6. Note that from inequality (5.12) by Gonshor transfer one obtains

(5.38)

Substitution Equation (5.21) into Equation (5.36) gives

(5.39)

Multiplying Equation (5.39) by Wattenberg hyperinteger by Theorem 2.13 (see sub- section 2.8) we obtain

(5.40)

By using inequality (5.38) for a given we will choose infinite prime integer such that:

(5.41)

Therefore from Equations (5.40) and (5.41) by using definition (2.15) of the function given by Equation (2.20)-Equation (2.21) and corresponding basic property I (see Subsection 2.7) of the function we obtain

(5.42)

From Equation (5.42) finally we obtain the main equality

(5.43)

We will choose now infinite prime integer in Equation (3.56) such that

(5.44)

Hence from Equation (5.16) follows

(5.45)

Note that Using (5.44) and (5.45) one obtains:

(5.46)

Using Equation (5.11) one obtains

(5.47)

Part IV. The proof of the inconsistency of the main equality (5.43)

In this subsection we wil prove that main equality (5.43) is inconsistent. This proof is based on the Theorem 2.10 (v), see Subsection 2.6.

Lemma 5.3. The equality (5.43) under conditions (5.46)-(5.47) is inconsistent.

Proof. (I) Let us rewrite Equation (5.43) in the short form

(5.48)

where

(5.49)

From (5.46)-(5.47) follows that

(5.50)

Remark 5.7. Note that Otherwise we obtain that

(5.51)

But the other hand from Equation (5.48) follows that

(5.52)

But this is a contradiction. This contradiction completed the proof of the statement (I).

(II) Let and be the external sum correspondingly

(5.53)

Note that from Equation (5.43) and Equation (5.53) follows that

(5.54)

Lemma 5.4. Under conditions (5.46)-(5.47)

(5.55)

and

(5.56)

Proof. First note that under conditions (5.46)-(5.47) one obtains

(5.57)

Suppose that there exists such that hen from Equation (5.54) one obtains

(5.58)

From Equation (5.58) by Theorem 2.17 one obtains

(5.59)

Thus

(5.60)

From Equation (5.60) by Theorem 2.11 follows that and therefore by Lemma 5.2 one obtains the contradiction. This contradiction finalized the proof of the Lemma 5.4.

(III)

Remark 5.8. (i) Note that from Equation (5.49) by Theorem 2.10 (v) follws that has the form

(5.61)

where

(5.62)

(ii) Substitution by Equation (5.61) into Equation (5.48) gives

(5.63)

Remark 5.9. Note that from (5.63) by definitions follows that

(5.64)

Remark 5.10. Note that from (5.62) by construction of the Wattenberg integer obviously follows that there exists some such that

(5.65)

Therefore

(5.66)

Note that under conditions (5.46)-(5.47) and (5.66) obviously one obtains

(5.67)

From Equation (5.63) follows that

(5.68)

Therefore

(5.69)

From (5.69) follows that

(5.70)

Note that from (5.70) by Theorem 2.8 (see Subsection 2.5) and Formula (5.32) one otains

(5.71)

From Equation (5.70)-Equation (5.71) follows that

(5.72)

Thus

(5.73)

and therefore

(5.74)

But this is a contradiction. This contradiction completed the proof of the Lemma 5.3.

Remark 5.11. Note that by Definition 2.18 and Theorem 2.18 from Assumption 5.1 and Assumption 5.2 follows

Theorem 5.3.The equality (5.75) is inconsistent.

Proof. The proof of the Theorem 5.3 obviously copies in main details the proof of the Theorem 5.3.

Theorem 5.3 completed the proof of the main Theorem 1.6.

Cite this paper

JaykovFoukzon, (2015) Non-Archimedean Analysis on the Extended Hyperreal Line *Rd and the Solution of Some Very Old Transcendence Conjectures over the Field Q. Advances in Pure Mathematics,05,587-628. doi: 10.4236/apm.2015.510056

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