The Wonders of Arithmetic from Pierre Simon de Fermat

Youri Veniaminovich Kraskov, 2021

This book shows how the famous scientific problem called "Fermat Last theorem" (FLT) allows us to reveal the insolvency and incapacity of science, in which arithmetic for various historical reasons has lost the status of the primary basis of all knowledge. The unusual genre of the book was called "Scientific Blockbuster", what means a combination of an action-packed narrative in the style of fiction with individual fragments of purely scientific content. The original Russian text of this book is translated into English by its author Youri Kraskov.

3. What is a Number?

3.1. Definition the Notion of Number

The question about the essence the notion of number at all times was for scientists the thing-in-itself. They of course, understood that they could not distinctly answer this question as well as they could not admit in this since this would have a bad effect on maintaining the prestige of science. What is the problem here? The fact is that in all cases a number must be obtained from other numbers, otherwise it cannot be perceived as a number. To understand for example, the number 365, you need to add three hundred with six tens and five units. It follows that the notion of a number does not decompose into components that are qualitatively different from it and in such a way as usual for science i.e. through analysis, it is not possible to penetrate the secret of its essence.

Scientists having a question about the nature of numbers immediately ran into this problem and came to the conclusion that a general definition the notion of number simply does not exist. But not a such was Pierre Fermat who approached this problem from other side. He asked: “Where does the notion of number come from?” And came to the conclusion that his predecessors were the notions “more”, “less” and “equal” as the comparisons’ results of some properties inherent to different objects [30].

If different objects are compared in some property with the same object then such a notion as a measurement appears, so perhaps is the essence of a number possible revealed through a measurement? However, it is not so. In relation to the measurement, the number is primary i.e. if there are no numbers, there can be no also measurements. Understanding the essence of the number becomes possible only after establishing the number is inextricably connect with the notion of “function”.

But this notion is not difficult to determine:

A function is a given sequence of actions with its arguments.

In turn, actions cannot exist on their own i.e. in the composition of the function in addition to them must include the components, with which these actions are performed. These components are called function arguments. From here follows a general definition the notion of number:

Number is an objective reality existing as a countable quantity, which consists of function arguments and actions between them.

For example, a+b+c=d where a, b, c are arguments, d is a countable quantity or the number value.³²

To understand what a gap separates Pierre Fermat from the rest of the science’s world, it is enough to compare this simple definition with the understanding existing in today's science [13, 29]. But understanding clearly presenting in the scientific works of Fermat, allowed him still in those distant times to achieve results that for other scientists were either fraught with extreme difficulties or even unattainable. It may be given also the broader definition the notion of number, namely:

A number is a kind of data represented as a function.

This extended definition the notion of number goes beyond frameworks mathematics; therefore, it can be called as general one and the previous definition as mathematical. In this second definition, it is necessary to clarify the essence the notion of “data”, however, for modern science this question is no less difficult than the question about the essence of the notion a number.³³

From the general definition the notion of number follows the truth of the famous Pythagoras' statement that everything existing can be reflected as a number. Indeed, if a number is a special kind of information, this statement very bold at that time, was not only justified, but also confirmed by the modern practice of its use on computers where three well-known methods of representing data are implemented: numerical (or digitized), symbolic (or textual) and analog (images, sound, and video). All three methods exist simultaneously.

Pic. 30. Pythagoras

A strikingly bold statement even for our time that thinking is an unconscious process of computations, have been expressed in the 17th century by Gottfried Leibniz. Here, thinking is obviously understood as the process of data processing, which in all cases can be represented as numbers. Then it is clear how computations appear, but understanding of the essence of this process in modern science is so far lacking.³⁴

Pic. 31. Gottfried Leibniz

All definitions of a number have one common basis:

Numbers exist objectively in the sense that they are present in the laws of the world around us, which can be known only through numbers.

From the school bench everyone will learn about numbers from the childish counting: one, two, three, four, five etc. Only the Lord knows where did this counting come from. However, there were attempts to explain its origin using axioms, but the origin of them is as incomprehensible as the counting. Rather, it looks like a certain imitation of the Euclid's"Elements"to add to knowledge the image of science and the appearance of solidity and fundamentality.

The situation is completely different when there is a mathematical definition the essence of a number. Then for a more complete understanding of it, both axioms and a countable quantity become a necessity. Indeed, this definition to the essence of a number includes arguments, actions and a countable quantity. But arguments are also numbers and they should be presented not specifically each of them, but by default i.e. in the form of a generally accepted and unchanged function, which is called the number system, however it no way could to appear without such a notion as a count. Now, axioms turn out to be very appositely and without them a count may be got only from aliens. In reality it was namely so happened since such sources of knowledge as the Euclid's “Elements” or the Diophantus' “Arithmetic” were clearly created not by our, but by a completely different civilization.³⁵

If axioms regulate the count, then they are primary in relation to it. However, there is no need to determine their essence through the introduction of new notions because the meaning of any axioms is precisely in their primacy i.e., they are always essentially the boundaries of knowledge. Thus, axioms receive an even more fundamental status, than until now when they were limited only to the foundation of any separate system. In particular, the system of axioms, developed by the Italian mathematician Giuseppe Peano, very closely correspond to the solution of the problem for constructing a counting system although this main purpose was not explained apparently with a hint on justification the essence the notion of number. The scientific community perceived them only as a kind of “formalization of arithmetic” completely not noticing that these axioms in no way reflect the essence of numbers, but only create the basis for their presentation by default i.e. through a count.

If the main content of axioms is to determine the boundaries of knowledge related to generally accepted methods of representing of numbers, then they should be built both from the definition the essence of the notion of number and in order to ensure the strength and stability of the whole science's building. Until now, due to the lack of such an understanding of the ways of building the foundations of knowledge, the question about the essence of numbers has never even been asked, but only complicated and confused.

Pic. 32. Giuseppe Peano

However, now when it becomes clearer and without any special difficulties, all science can receive a new and very powerful impetus for its development. And then namely on such a solid basis, science acquires the ability to overcome with an incredible ease such complex obstacles, which in the old days, when there was no understanding the essence of numbers, they seemed to science as completely impregnable fortresses³⁶.

3.2. Axioms of Arithmetic

3.2.1. Axioms of a Count

This path was first paved at the end of the 19th century by Peano axioms.³⁷ We will make changes to them based on our understanding the essence of the number.

Axiom 1. A number is natural if it is added of units.³⁸

Axiom 2. The unit is the initial natural number.

Axiom 3. All natural numbers form an infinite row, in which each

following number is formed by adding unit to the previous number.

Axiom 4. The unit does not follow any natural number.

Axiom 5. If some proposition is proven for unit (the beginning of

induction) and if from the assumption that it is true for a natural

number N, it follows that it is also true for a natural number

following N (induction hypothesis), then this sentence will be true

for all natural numbers.

Axiom 6. In addition to natural numbers, there can exist another

numbers derived from them, but only in the case if they possess all

without exception the basic properties of natural numbers.

The first axiom is a direct consequence from definition the essence of number, so Peano simply could not have it. Now this first axiom conveys the meaning of defining the notion of number to all another axiom. The second, fourth, and fifth axioms are preserved as in Peano version almost unchanged, but the fourth axiom of Peano is completely removed from this new system as redundant.

The second axiom has the same meaning as the first one in the Peano list, but is being specified in order to become a consequence of the new first axiom.

The third axiom is the new wording of Peano's second axiom. The notion of the natural row is given here more simply than by Peano where you need to guess about it through the notion of the “next” number. The fourth axiom is exactly the same as the third axiom of Peano.

The fifth axiom is the same as by Peano, which is considered the main result of the entire system. In fact, this axiom is the formulation the method of induction, which is very valuable for science and in this case allows to justify and build a count system. However, a count is present in one or another form not only in natural numbers, but also in any other numbers, therefore one more final axiom is needed.

The sixth axiom extends the basic properties of natural numbers to any numbers derived from them because if it turns out that any quantities obtained by calculations from natural numbers, contradict their basic properties, then these quantities cannot belong to the category of numbers.

Now arithmetic gets all the prerequisites in order to have the status the most fundamental of all scientific disciplines. From the point of view the essence of a count everything becomes much simpler and more understandable than until now. On the basis of this updated system of axioms there is no need to “create” natural numbers one after another and then “prove” the action of addition and multiplication for the initial numbers. Now it’s enough just to give names to these initial numbers within the framework of the generally accepted number system.

If this system is decimal, then the symbols from 0 to 9 should receive the status of the initial numbers composed of units in particular: the number “one” is denoted as 1=1, the number “two” is denoted as 2=1+1, the number “ three ” as 3=1+1+1 etc. up to the number nine. Numbers after 9 and up to 99 adding up from tens and ones for example, 23=(10+10)+(1+1+1) and get the corresponding names: ten, eleven, twelve… ninety-nine. Numbers after 99 are made up of hundreds, tens and units, etc. Thus, the names of only the initial numbers must be preliminarily counted from units. All other numbers are named so that their quantity can be counted using only the initial numbers.³⁹

3.2.2. Axioms of Actions

All arithmetic actions are components of the definition the essence of the number. In a compact form they are presented as follows:

1. Addition: n=(1+1…)+(1+1+1…)=(1+1+1+1+1…)

2. Multiplication: a+a+a+…+a=a×b=c

3. Exponentiation: a×a×a×…×a=a^b=c

4. Subtraction: a+b=c → b=c−a

5. Division: a×b=c → b=c:a

6. Logarithm: a^b=c → b=log_ac

Hence, necessary definitions can be formulated in the form of axioms.

Axiom 1. The action of adding several numbers (summands) is their

association into one number (sum).

Axiom 2. All arithmetic actions are either addition or derived from

addition.

Axiom 3. There are direct and inverse arithmetic actions.

Axiom 4. Direct actions are varieties of addition. Besides the addition

itself, to them also relate multiplication and exponentiation.

Axiom 5. Inverse actions are the calculation of function arguments.

These include subtraction, division and logarithm.

Axiom 6. There aren’t any other actions with numbers except for

combinations of six arithmetic actions.⁴⁰

3.2.3. Basic Properties of Numbers

The consequence to the axioms of actions are the following basic properties of numbers due to the need for practical calculations:

1. Filling: a+1>a

2. The neutrality of the unit: a×1=a:1=a

3. Commutativity: a+b=b+a; ab=ba

4. Associativity: (a+b)+c=a+(b+c); (ab)c=a(bc)

5. Distributivity: (a+b)c=ac+bc

6. Conjugation: a=c → a±b=b±c; ab=bc; a:b=c:b; ab=cb; log_ba=log_bc

These properties have long been known as the basics of primary school and so far, they have been perceived as elementary and obvious. The lack of a proper understanding of the origin of these properties from the essence the notion of number has led to the destruction of science as a holistic system of knowledge, which must now be rebuilt beginning from the basics and preserving herewith everything valuable that remains from real science.

The presented above axiomatics proceeds from the definition the essence the notion of number and therefore represents a single whole. However, this is not enough to protect science from another misfortune i.e. so that in the process of development it does not drown in the ocean of its own researches or does not get entangled in the complex interweaving of a great plurality of different ideas.

In this sense, it must be very clearly understood that axioms are not statements accepted without proof. Unlike theorems, they are only statements and limitations synthesized from the experience of computing, without of which they simply cannot be dispensed. Another meaning is in the basic theorems, which are close to axioms, but provable. One of them is the Basic or Fundamental theorem of arithmetic. This is such an important theorem that its proof must be as reliable as possible, otherwise the consequences may be unpredictable.

Pic. 33. Initial Numbers Pyramids

3.3. The Basic Theorem of Arithmetic

3.3.1. Mistakes of the Greats and the Fermat's Letter-Testament

The earliest known version of the theorem is given in the Euclid's"Elements"Book IX, Proposition 14.

If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it.

The explain is following: “Let the number A be the least measured by the prime numbers B, C, D. I say that A will not be measured by any other prime number except B, C, D”. The proof of this theorem looks convincing only at first glance and this visibility of solidity is strengthened by a chain of references: IX-14 → VII-30 → VII-20 → VII-4 → VII-2.

However, an elementary and even very gross mistake was made here. Its essence is as follows:

Let A=BCD where the numbers B, C, D are primes. If we now assume the existence of a prime E different from B, C, D and such that A=EI then we conclude that in this case A=BCD is not divisible by E.

This last statement is not true because the theorem has not yet been proven and it doesn’t exclude for example, BCD=EFGH where E, F, G, H are primes other than B, C, D. Then

A:E=BCD:E=EFGH:E=FGH

i.e. in this case it becomes possible that the number A can be divided by the number E and then the proof of the theorem is based on an argument that has not yet been proven, therefore, the final conclusion is wrong. The same error can take place also in other theorems using decomposition of integers into prime factors. Apparently, due to the archaic vocabulary Euclid's “Elements” even such a great scientist as Euler did not pay due attention to this theorem, otherwise, he would hardly have begun to use “complex numbers” in practice that are not subordinate to it.

The same story happened with Gauss who also did not notice this theorem in the Euclid's"Elements", but nevertheless, formulated it when a need arose. The formulation and proof of Gauss are follows:

“Each compose number can be decomposed into prime factors in a one only way.

If we assume that a composite number A equal to a^αb^βc^γ…, where a, b, c,… denote different primes, can be decomposed into prime factors in another way, then it is first of all clear that in this second system of factors, there cannot be other primes except a, b, c,…, because the number A composed of these latter cannot be divisible by any other prime number” [11, 25].

This is an almost exact repeating of erroneous argument in the Euclid's proof. But if this theorem is not proven, then the whole foundation of science built on natural numbers collapses and all the consequences of the definitions and axioms lose their significance. And what to do now? If such giants of science as Euclid and Gauss could not cope with the proof of this theorem, then what we sinners can to do. But yet there is a way out and it is indicated in one amazing document called"Fermat's Letter-Testament".

This letter was sent by Fermat in August 1659 to his longtime friend and former colleague in the Parliament of Toulouse the royal librarian Pierre de Carcavy from whom he was received by the famous French scientist Christian Huygens who was the first to head the French Academy of Sciences created in 1666. Here we give only some excerpts from this Fermat's letter, which are of particular interest to us [9, 36].

“Summary of discoveries in the science about numbers.…

1. Since the usual methods set in the Books are not sufficient to prove very difficult sentences, I finally found a completely special way to solve them. I called this method of proof infinite or indefinite descent. At first, I used it only to prove negative sentences such as:… that there exists no a right triangle in numbers whose area is a square”. See Appendix II for details.

The science about numbers is called here arithmetic and the further content of the letter leaves no doubt about it. Namely with arithmetic not only mathematical, but also all other sciences begin. In arithmetic itself the descent method is one of the fundamental one. The following are examples of problems whose solution without this method is not only very difficult, but sometimes even hardly to be possible. Here we will name only a few of these examples.

"2. For a long time, I could not apply my method to affirmative sentences because rounds and circuitous ways to achieve the aim are much more difficult than those that served me for negative sentences. Therefore, when I needed to prove that every prime number that is by unit more than multiple of four, consists of <sum of> two squares, I was in a greatest difficulty. But finally, my thoughts repeated many times shed light that I did not have and the affirmative sentence became possible to interpret with my method using some new principles that needed to be attached to them. This progress in my reasoning for the case of affirmative sentences is as follows: if some prime number that on 1 exceeds the multiplied of 4, does not consist of two squares, then there is another a prime number of the same nature, smaller than this and then a third, also smaller etc. going down until you come to the number 5, which is the smallest from all numbers of this nature. It therefore, cannot consist of two squares, what however, takes place. From this by proof from the contrary we can conclude that all primes of this nature should consist of two squares”.

This Fermat’s theorem was first proven by Euler in 1760 [6, 38], (see Appendix III), and in the framework of the very complex Gauss'"Deductive Arithmetic"this theorem is proving in one sentence [23]. However, no one succeeded in repeating the proof of Fermat himself.

“… 3. There are infinitely many questions of this kind, but there are others that require new principles for applying the descent method to them… This is the next question that Bachet as he confesses in his commentary on Diophantus, could not prove. On this occasion, Descartes made the same statement in his letters acknowledging that he considers it so difficult that he sees no way to solve it. Each number is a square or consists of two, three or four squares".

Else earlier, 22 years ago, in October 1636 in a letter to Mersenne Fermat reported on the same problem as about his discovery, but in general form i.e. for any polygonal numbers (for example, triangles, squares, pentagons etc.). Subsequently, he even called this theorem golden one. Consequently, the method of descent was discovered by him at the very beginning of his research on arithmetic. By the time of writing the letter-testament, Fermat already knew from Carcavy that the question of foundation the French Academy of Sciences was practically resolved and he needed only to wait for the building to be completed, so it come true his life's dream to become a professional scientist in the rank of academician. Huygens was commissioned to collect materials for the first academic publications. Fermat proposed for them the method of descent discovered by him and the solution of specific arithmetic tasks on its basis.

However, only few people knew that these tasks were very difficult and Fermat understood that if he would publish their solutions, they would not make any impression at all. He already had such an experience and now he has prepared a real surprise. For those who don't appreciate the value of his solution, he would offer to solve another task. This is the Basic theorem of arithmetic, which is of particular importance for all science since without it the whole theory loses its strength. Fermat found a mistake in the proof of Euclid and came to the conclusion that to prove this theorem without applying the descent method is extremely difficult if at all possible. However, now we can also reveal this secret with the help of our opportunities to look into Fermat’s cache with “heretical writings” and return his lost proof to science in the form of the reconstruction presented below.

3.3.2. The Proof of Fermat

So, to prove the Basic theorem of arithmetic we suppose that there exist equal natural numbers A, B consisting of different prime factors:

A=B (1)

where A=pp₁p₂ …p_n; B=хx₁x₂ …x_m ; n≥1; m≥1

Due to the equality of the numbers A, B each of them is divided into any of the prime numbers p_i or x_i. Each of the numbers A, B can consist of any set of prime factors including the same ones, but at the same time there is no one p_i equal to x_i among them, otherwise they would be in (1) reduced. Now (1) can be represented as:

pQ=xY (2)

where p, x are the minimal primes among p_i, x_i; Q=A/p; Y=B/x.

Since the factors p and x are different, we agree that p>x; x=p–δ₁ then

pQ=(p — δ₁)(Q+δ₂) (3)

where δ₁=p–x; δ₂=Y–Q

From (3) it follows that Qδ₁=(p — δ₁)δ₂ or

Qδ₁=xδ₂ (4)

Equation (4) is a direct consequence of assumption (1). The right side of this equation explicitly contains the prime factor x. However, on the left side of equation (4) the number δ₁ cannot contain the factor x because δ₁ = p — x is not divisible by x due to p is a prime. The number Q also does not contain the factor x because by our assumption it consists of factors p_i among which there is not a single equal to x. Thus, there is a factor x on the right in equation (4), but not on the left. Nevertheless, there is no reason to argue that this is impossible because we initially assume the existence of equal numbers with different prime factors.

Then it remains only to admit that if there exist natural numbers A = B composed of different prime factors, then it is necessary that in this case there exist another natural number A₁=Qδ₁ and B₁=xδ₂; also equal to each other and made up of different prime factors. Given that δ₁=(p–x)<p, and δ₂=(Y–Q)<Y and also, after comparing equation (4) with equation (2), we can state:

A₁ = B₁, where A₁<A; B₁<B (5)

Now we get a situation similar to the one with numbers A, B only with smaller numbers A₁, B₁. Analyzing now (5) in the manner described above we will be forced to admit that there must exist numbers

A₂=B₂, where A₂<A₁; B₂<B1 (6)

Following this path, we will inevitably come to the case when the existence of numbers A_k=B_k, where A_k<A_k-1; B_k<B_k-1 as a direct consequence of assumption (1) will become impossible. Therefore, our initial assumption (1) is also impossible and thus the theorem is proven.⁴¹

Looking at this very simple and even elementary proof by the descent method naturally a puzzling question arise, how could it happen that for many centuries science not only had not received this proof, but was completely ignorant that it had not any one in general? On the other hand, even being mistaken in this matter i.e. assuming that this theorem was proven by Euclid, how could science ignore it by using the"complex numbers"and thereby dooming itself to destruction from within? And finally, how can one explain that this very simple in essence theorem, on which the all science holds, is not taught at all in a secondary school?

As for the descent method, this proof is one of the simplest examples of its application, which is quite rare due to the wide universality of this method. More often, the application of the descent method requires a great strain of thought to bring a logical chain of reasoning under it. From this point of view, some other special examples of solving problems by this method can be instructive.

3.4. The Descent Method

3.4.1. A Little Bit of"Sharpness of Mind"for a Very Difficult Task

We will now consider another example of the problem from Fermat's letter-testament, which is formulated there as follows:

There is only one integer square, which increased by two, gives a cube, this square is 25.

When at the suggestion of Fermat, the best English mathematician of the time John Wallis tried to solve it, he was very vexed and forced to acknowledge he could not do it. For more than two centuries it was believed that Leonard Euler received the solution to this problem, but his proof is based on the use of"complex numbers", while we know these are not numbers at all because they do not obey the Basic theorem of arithmetic. And only at the end of the twentieth century André Weil using the Fermat's triangles method still managed to get a proof [17].

It was a big progress because a purely arithmetic method was used here, however, as applied to this problem, it was clearly dragged the ears. Could Fermat solve this problem easier? We will also extract the answer to this question from the cache, what will allow us to reveal this secret of science in the form of the following reconstruction. So, we have the equation p³=q²+2 with the obvious solution p=3, q=5. To prove Fermat's assertion, we suppose that there is another solution P>p=3, Q>q=5, which satisfies the equation

P³=Q²+2 (1)

Since it is obvious that Q>P then let Q=P+δ (2)

Substituting (2) in (1) we obtain: P²(P–1)–2δP–δ²=2 (3)

Here we need just a little bit of “sharpness of mind” to notice that δ>P otherwise equation (3) is impossible. Indeed, if we make a try δ=P then on the left (3) there will be P²(P–4)>2 what is not suitable, therefore there must exist a number δ₁=δ–P. Then substituting δ=P+δ₁ in (3) we obtain

P₂(P–4)–4δ₁P–δ₁²=2 (4)

Now we will certainly notice that δ₁>P otherwise, by the same logic as above, on the left (4) we get P²(P–9)>2 what again does not suitable, then there must exist a number δ₂=δ₁–P and after substituting δ₁=P+δ₂ in (4), we obtain P²(P–9)–6δ₂P–δ₂²=2 (5)

Here one can no longer doubt that this will continue without end. Indeed, by trying δ_i=P each time we get P²(P−K_i)>2. Whatever the number of K_i this equation is impossible because if K_i<P and P>3 then P²(P−K_i)>2 and if K_i≥P then this option is excluded because then P²(P−K_i)≤0

To continue so infinitely is clearly pointless, therefore our initial assumption of the existence of another solutions P>3, Q>5 is false and this Fermat's theorem is proven.

In the book of Singh, which we often mention, this task is given as an example of the “puzzles” that Fermat was “inventing”. But now it turns out that the universal descent method and a simple technique with trying, make this task one of the very effective examples for learning at school.

Along with this proof, students can easily prove yet another theorem from Fermat’s letter-testament, which could be solved only by such a world-famous scientist as Leonard Euler:

There are only two squares that increased by 4, give cubes, these squares will be 4 and 121.

In other words, the equation p³=q²+4 has only two integer solutions.

3.4.2. The Fermat’s Golden Theorem

We remind that in the Fermat's letter-testament only a special case of this theorem for squares is stated. But also, this simplified version of the task was beyond the power not only of representatives of the highest aristocracy Bachet and Descartes, but even the royal-imperial mathematician Euler.

However, another royal mathematician Lagrange, thanks to the identity found by Euler, still managed to cope with the squares and his proof of only one particular case of FGT is still replicated in almost all textbooks. However, there is no reasonable explanation that the general proof of the FGT for all polygonal numbers obtained by Cauchy in 1815 was simply ignored by the scientific community.

We begin our study with the formulation of the FGT from Fermat's letter to Mersenne in 1636. It is presented there as follows:

Every <natural> number is equal

one, two or three triangles,

one, 2, 3 or 4 squares,

one, 2, 3, 4 or 5 pentagons,

one, 2, 3, 4, 5 or 6 hexagons,

one, 2, 3, 4, 5, 6 or 7 heptagons,

and so on to infinity [36].

Since polygonal numbers are clearly not respected by today's science, we will give here all the necessary explanations. The formula for calculating any polygonal number is represented as

m_i = i+(k−2)(i−1)i/2

where m is a polygonal number, i is a serial number, k is the quantity of angles.

Thus, m₁=1; m₂=k; and for all other i the meaning of mi varies widely as shown in the following table:

Table 1. Polygonal numbers

To calculate m_i it is enough to obtain only triangular numbers by the formula, which is very easily since the difference between them grows by unit with each step. And all other m_i can be calculated by adding the previous triangular number in the columns. For example, in column i=2, numbers increase by one, in column i=3 — by three, in column i=4 — by six etc. i.e. just on the value of the triangular number from the previous column.

To make sure that any natural number is represented by the sum of no more than k k-angle numbers is quite easily. For example, the triangular number 10 consists of one summand. Further 11=10+1, 12=6+6, 13=10+3 of two, 14=10+3+1 of three, 15 again of one summand. And so, it will happen regularly with all natural numbers. Surprisingly that the number of necessary summands is limited precisely by the number k. So, what is this miraculous power that invariably gives such a result?

As an example, we take a natural number 41. If as the summand triangular number will be closest to it 36, then it will not in any way to fit into three polygonal numbers since it consists minimum of 4 ones i.e. 41=36+3+1+1. However, if instead of 36 we take other triangular numbers for example, 41=28+10+3, or 41=21+10+10 then again in some unknown miraculous way everything will so as it stated in the FGT.

At first glance it seems simply unbelievable that it can somehow be explained? But we still pay attention to the existence of specific natural numbers, which are consisting at least of k k-angle numbers and denoted by us as S-numbers. Such numbers are easily to find for example, for triangles — 5, 8, 14, for squares — 7, 15, 23, for pentagons — 9, 16, 31 etc. And this our simple observation allows us directly to move to aim i.e. without using ingenious tricks or powerful"sharpness of mind".

Now to prove the FGT, suppose the opposite i.e. that there exists a certain minimal positive integer N consisting minimum of k + 1 k-angle numbers. Then it’s clear that this our supposed number should be between some k-angle numbers m_i and m_i+1 and can be represented as

N=m_i+δ₁ where δ₁=N−m_i (1)

It is quite obvious that δ₁ must be an S-number since otherwise this would contradict our assumption about the number N. Then we proceed the same way as in our example with the number 41 i.e. represent the supposed number as

N = m_i-1+δ₂ where δ₂=N−m_i-1

Now δ₂ should also be an S-number. And here so we will go down to the very end i.e. before

δ_i-1=N−m₂ =N−k and δ_i=N−m₁=N–1 (2)

Thus, in a sequence of numbers from δ₁ to δ_i, all of them must be S-numbers i.e. each of them will consist of a sum minimum of k k-angle numbers, while our supposed number N will consist minimum of k+1 k-angle numbers. From (1) and (2) it follows:

N− m_i =S_i (3)

Thus, if we subtract any smaller polygonal number m_i from our supposed number N then according to our assumption, the result should be only an S-number. Of course, this condition looks simply unbelievable and it seems that we are already at target, but then how can one prove that this is impossible?…

If we gave an answer to this question here, then this famous Fermat's theorem would immediately turn into the most common school problem and interest to it would be lost. To prevent this from happening, we will stay on the fact that the proof is presented here only by 99% and the remaining 1% will be offered to those who will be interested in order to appreciate the true magnificence of this scientific achievement of Fermat, especially in comparison with the Cauchy’s GFT proof.⁴²

Pic. 34. Title Page the Cauchy's Proof

of the Fermat's Golden Theorem

Pic. 35. One of 43 Pages the Cauchy's Proof

of the Fermat's Golden Theorem

3.4.3. Archimedes-Fermat Problem

The problem statement is as follows:

Let any non-square number be given, you need to find an infinite number of squares, which after multiplication by this number and increasing by unit, will make a square.

Fermat proposed finding solutions for the numbers 61, 109, 149, and 433 [36].

The English mathematician John Wallis managed to find a way to calculate the required numbers using the Euclidean method of decomposing an irrational number into an infinite common fraction. He published his decision under the name"Commercium epistolicum"see pic. 37-38.

Pic. 36. John Wallis

Pic. 37. Title Page of Wallis's Publication Commercium Epistolicum

Although Wallis did not give a complete proof the validity of this method, Fermat nevertheless admitted that he had coped with the task. Euler came very close to the solution when he showed that this fraction is cyclical, but he was not able to complete the proof and this task was finally solved by Lagrange. Later, this Fermat's task also was solved by Gauss in his own way, but for this purpose the extensive theory he created called “Arithmetic of deductions” was involved. And everything would be fine if the Lagrange's proof was not in the category of highest difficulty and the Gauss decision was not based on the most complicated theory. Fermat himself clearly could not follow such ways. About how he himself solved this problem, he reports in the letter-testament to Carcavy in August 1659 [36]: “I recognize that Mr. Frenicle gave various special solutions to this question as well as Mr. Wallis, but a common solution will be found using the method of descent applied skillfully and appropriately."However, this Fermat's solutions so remained as the secret behind seven seals!

Pic. 38. Page 64 Commercium Epistolicum

Demonstrating Wallis Method

We will try here slightly to open the veil over this mystery. To do this, we will look at a simple example of Wallis calculations and then compare it with how one could do these calculations using Fermat's method. So, we need to find the smallest numbers x and y that satisfy the equation Ax² + 1 = y². Let A = 29 then calculations by the Wallis method look as follows [32]:

From this sequence of calculations, a chain of suitable fractions is obtained by backward i.e. from a5 to a0 and looks like: 5/1; 11/2; 16/3; 27/5. As a result, we get 70/13. Then the minimum solution would be:

x₁√29+у₁=(13√29+70)²=1820√29+9801; x₁=1820; y₁=9820

Wallis was unable to prove that this method of computation gives solutions for any non-square number A. However, he guessed that the chain of computations ends where a6 will be computed by the same formula as a1. To understand the meaning of this chain of calculations, you need to study a very voluminous and extremely difficult theory [7, 14, 19, 23, 26, 32], which Fermat could not have developed at that time. Since no Fermat's manuscripts on arithmetic have survived, a natural question arises: how could he formulate such a difficult problem, about which there was very little information before him?

For today's science such a question is clearly beyond its capabilities since for it the pinnacle of achievements in solving Fermat's problems is any result even inflated to such incredible dimensions that we have today. However, it is difficult to imagine how much this our respected science will be dejected when from this book it learns that the problem was solved by Fermat not for great scientists, but… for schoolchildren!!! However, here we cannot afford to grieve science so much, so we only note that the example given in the textbooks is very unfortunate since it can be solved quite simply, namely: x = 2mz, where m<x, z<y, Am²−1 = z². This last equation differs from the initial one only in sign and even by the method of ordinary tests without resorting to irrational numbers one can easily find the solution m = 13; z = 70; x = 2 x 13 x 70 = 1820; y = 9820.

Obviously, in textbooks it would be much more appropriate to demonstrate an example with the number 61 i.e. the smallest number proposed by Fermat himself. How he himself solved this problem is unknown to science, but we have already repeatedly demonstrated that it is not a problem for us to find out. We just need to look once more into the cache of the Toulousean senator and as soon as we succeeded, we quickly found the right example so that it could be compared with the Wallis method. In this example you can calculate x = 2mz, where m and z are solutions to the corresponding equation 61m² — z² = 1. Then the chain of calculations is obtained as follows:

61m²−z²=1

m=(8m₁±z₁)/3=(8×722+5639)/3=3805; z²=61×3805²−1=29718²

61m₁²−z₁²=3

m=(8m₁±z₁)/3=(8×722+5639)/3=3805; z₁²=61×722²−1=29718²

61m₂²−z₂²=9

m=(8m₁±z₁)/3=(8×722+5639)/3=3805; z₂²=61×137²−1=29718²

61m₃²−z₃²=27

m₃=(8m₄±z₄)/3=(8×5+38)/3=26; z₃²=61×26²−27=203²

61m₄²−z₄²=81

m₄=(8m₅±z₅)/3=(8×2−1)/3=5; z₄²=61×5²−81=38²

61m₅²−z₅²=243

m₅=2; z₅²=1

We will not reveal all nuances of this method, otherwise all interest to this problem would have been lost. We only note that in comparison with Wallis method where the descent method is not used, here it is present in an explicit form. This is expressed in the fact that if the numbers m and z satisfying the equation 61m²–z²=1 exist, then there must still exist numbers m₁<m and z₁<z satisfying the equation 61m₁²–z₁²=3, as well as the numbers m₂<m₁ and z₂<z₁, from equation 61m₂²–z₂²=9, etc. up to the minimum values m₅<m₄ and z₅<z₄. The number 3 appearing in the descent is calculated as 64 — 61, that is, as the difference between 61 and the square closest to it. Calculations as well as in the Wallis method are carried out in the reverse order i.e. only after the minimum values of m₅ and z₅ have been calculated. As a result, we get:

m=3805; z=29718

x=2mz=2×3805×29718=226153980

y=√(61×2261539802+1)=1766319049

Of course, connoisseurs of the current theory will quickly notice in this example that the results of calculations obtained in it will exactly coincide with those that can be obtained by the Wallis' method. However, for this they will have to use the irrational number √61, and our example with Fermat's method showed that it is possible to do calculations exclusively in the framework of arithmetic i.e. only in natural numbers. There is no doubt also that experts without much effort will guess how to get the formulas shown in our example. However, it will not be easily for them to explain how to apply this Fermat's method in the general case because from our example it is not at all clear how it is possible to determine that the ultimate goal is to solve the equation 61m₅² — z₅² = 243 from which calculations should be performed with a countdown.

It would be simply excellent if today's science could explain Fermat's method in every detail, but even the ghostly hopes for this are not yet visible. It would be more realistic to expect that attempts will be made to refute this example as demonstration a method of solving the problem unknown to science. Nevertheless, science will have to reckon with the fact that this example is still the only one in history (!!!) confirmation of what Fermat said in his letter-testament. When this secret is fully revealed, then all skeptics will be put to shame and they will have no choice, but to recognize Fermat as greater than all the other greatest scientists because they were recognized as such mainly because they created theories so difficult for normal people to understand that they could only cause immense horror among students who now have to take the rap for such a science.⁴³

https://www.youtube.com/watch?v=wFz8W2HsjfQ

https://www.youtube.com/watch?v=cUytn2SZ1n4

https://www.youtube.com/watch?v=ZhVNOgaBStY

In this sense, the following example of solving a problem using the descent method will be particularly interesting because it was proposed in a letter from Fermat to Mersenne at the end of 1636, i.e. the age of this task is almost four centuries! Euler's proof [8] was incorrect due to the use of"complex numbers"in it. However, even the revised version of André Weil in 1983 [17] is too complex for schooling.

3.4.4. Fermat’s Problem with Age 385 years

In the original version in 1636 this task was formulated as follows:

Find two square-squares, which sum is equal to a square-square,

or two cubes, which sum is a cube.

This formulation was used by Fermat's opponents as the fact that Fermat had no proof of the FLT and limited himself to only these two special cases. However, the very name"The Fermat’s Last Theorem"appeared only after the publication of"Arithmetic"by Diophantus with Fermat's remarks in 1670 i.e. five years after his death. So, there is no any reason to assert that Fermat announced the FLT in 1637.

The first case for the fourth power we have presented in detail in Appendix II. As for the case for the third power, Fermat's own proof method restored by us below, will not leave any chances to the solutions of this problem of Euler and Weil to remain in history of science, since from the point of view of the simplicity and elegance of the author's solution this problem, they will become just unnecessary.

Now then, to prove that there are no two cubes whose sum is a cube, we use the simplest approach based on divisibility of numbers, what means that in the original equation

a³+b³ = c³ (1)

the numbers a, b, and c can be considered as coprime ones, i.e. they do not have common factors, but in general case this is not necessary, since if we prove that equation (1) cannot have solutions in any integers, including those with common factors, then we will prove that coprime numbers also cannot be solutions of the original equation. Then we assume that both sides of equation (1) in all cases must be divisible by the number c², then equation (1) can be represented as

c³ = c²(x+y) = a³+b³ (2)

In this case, it is easily to see that there is only one way to get solutions to equation (1) when the numbers c, x, y, and x+y are cubes, i.e.

с = x+y = p³+q³= z³; x = p³; y = q³ (3)

Then equation (1) must have the form:

(z³)³ = (z²)³(p³+q³) (4)

Thus, we found that if there are numbers a, b, and c that satisfy equation (1), then there must be numbers p<a, q<b, and z<c that satisfy equations (3)

p³+q³= z³

If we now apply the same approach to solving this equation, that we applied to solving equation (1), we will get the same equation, only with smaller numbers. However, since it is impossible to infinitely reduce natural numbers, it follows that equation (1) has no solutions in integers.

At first glance, we have received a very simple and quite convincing proof of the Fermat problem by the descent method, which no one has been able to obtain in such a simple way for 385 years, and we can only be happy about it. However, such a conclusion would be too hasty, since this proof is actually incorrect and can be refuted in the most unexpected way.

However, this refutation is so surprising that we will not disclose it here, because it opens the way not only for the simplest proof of the FLT, but also automatically allows to reduce it to a very simple proof of the Beal conjecture. The disclosure the method of refuting this proof would cause a real commotion in the scientific world, therefore we will include this mystery among our riddles (see Appendix V Pt. 41).

So, we have demonstrated here solving to Fermat's problems (only by descent method!):

1) The proof of the Basic theorem of arithmetic.

2) The proof of the Fermat's theorem on the unique solving the

equation p³ = q² + 2.

3) A way to prove Fermat's Golden Theorem.

4) A Fermat's way to solve the Archimedes-Fermat equation

Ax² + 1 = y².

5) The proof method of impossibility a³+b³=c³ in integers, which

opens a way to simplest proofs of the FLT and Beal conjecture.

6) A Fermat's proof his grandiose discovery about primes in the

form 4n + 1 = a² + b² which we have presented in another style in

Appendix IV, story Year 1680.

Over the past 350 (!!!) years after the publication of these problems by Fermat, whole existing science could not even dream of such a result!

3.5. Parity Method

Before we embarking on the topic"Fermat's Last Theorem"we note that this problem was not solved by Fermat himself using the descent method, otherwise in his FLT formulation there would be no mention of a"truly amazing proof", which certainly related to other methods. Therefore, to the above examples of the application of the descent method we will add our presentation of two methods unknown to today's science. The most curious of these is the parity method.

3.5.1. Defining Parity as a Number

The Basic theorem of arithmetic implies a simple, but very effective idea of defining parity as a number, which is formulated as follows:

The parity of a given number is the quantity of divisions this number by two without a remainder until the result of the division becomes odd.

Let's introduce the parity symbol with angle brackets. Then the expression ‹x› = y will mean:

the parity of the number x is equal to y. For example, the expression"the parity of the number forty is equal to three"can be represented as: ‹40›= 3. From this definition of parity, it follows:

— parity of an odd number is zero.

— parity of zero is infinitely large.

— any natural number n can be represented as n = 2^w (2N — 1)

where N is the base of a natural number, w is its parity.

3.5.2. Parity Law

Based on the above definition the parity, it can be stated that equal numbers have equal parity. In relation to any equation this provision refers to its sides and is absolutely necessary in order for it to have solutions in integers. From here follows the parity law for equations:

Any equation can have solutions in integers if and only if the parities of both its sides are equal.

The mathematical expression for the parity law is W_L = W_R where W_L and W_R are the parities of the left and right sides of the equation respectively. A distinctive feature of the parity law is that the equality of numbers cannot be judged by the equality of their parity, but if their parities are not equal, then this certainly means the inequality of numbers.

3.5.3. Parity Calculation Rules

Parity of a sum or difference two numbers a and b

If ‹a› < ‹b› then ‹a ± b› = ‹a›.

It follows in particular that the sum or difference of an even and an odd number always gives a number with parity zero. If ‹a› = ‹b› = x then either ‹a + b› = x + 1 wherein ‹a — b› > x + 1

or ‹a — b› = x + 1 wherein ‹a + b› > x + 1

These formulas are due to the fact that

‹(a + b) + (a — b)› = ‹2a› = ‹a› + 1

It follows that the sum or difference of two even or two odd numbers gives an even number.

Parity of a sum or difference two power number aⁿ and bⁿ

If ‹a› < ‹b› then ‹aⁿ ± bⁿ› = ‹aⁿ›. If ‹a› = ‹b› = x then

only for even n:

‹aⁿ — bⁿ› = ‹a — b›+ ‹a + b›+ x(n — 2) + ‹n› — 1

‹aⁿ + bⁿ› = xn + 1

only for odd n:

‹aⁿ ± bⁿ› = ‹a ± b› + x(n — 1)

When natural numbers multiplying, their parities are added up

‹ab› = ‹a› + ‹b›

When natural numbers dividing, their parities are subtracted

‹a: b› = ‹a› — ‹b›

When raising number to the power, its parity is multiplied

‹a^b› = ‹a› × b

When extracting the root in number, its parity is divided

‹ ^b√a› = ‹a›: b

3.6. Key Formula Method

To solve equations with many unknowns in integers, an approach is often used when one more equation is added to the original equation and the solution to the original is sought in a system of two equations. We call this second equation the key formula. Until now due to its simplicity, this method did not stand out from other methods, however we will show here how effective it is and clearly deserves special attention. First of all, we note an important feature of the method, which is that:

Key formula cannot be other as derived from the original equation.

If this feature of the method is not taken into account i.e. add to the original equation some other one, then in this case, instead of solving the original equation we will get only a result indicating the compatibility of these two equations. In particular, we can obtain not all solutions of the original equation, but only those that are limited by the second equation.

In the case when the second equation is derived from the initial one, the result will be exhaustive i.e. either all solutions or insolvability in integers of the original equation. For example, we take equation z³ = x² + y². To find all its solutions we proceed from the assumption that a prerequisite (key formula) should be z = a² + b² since the right-hand side of the original equation cannot be obtained otherwise than the product of numbers which are the sum of two squares. This is based on the fact that:

The product of numbers being the sum of two squares in all

cases gives a number also consisting the sum of two squares.

The converse is also true: if it is given a composite number being the sum of two squares then it cannot have prime factors that are not the sum of two squares. This is easily to make sure from the identity

(a²+b²)(c²+d²)=(ac+bd)²+(ad−bc)²=(ac−bd)²+(ad+bc)²

Then from (a²+b²)(a²+b²)=(aa+bb)²+(ab−ba)²=(a²−b²)²+(ab+ba)² it follows that the square of a number consisting the sum of two squares, gives not two decompositions into the sum of two squares (as it should be in accordance with the identity), but only one, since (ab−ba)2= 0 what is not a natural number, otherwise any square number after adding to it zero could be formally considered the sum of two squares.

However, this is not the case since there are numbers that cannot be the sum of two squares.

As Pierre Fermat has established, such are all numbers containing at least one prime factor of type 4n − 1. Now from

a²−b²=c; ab+ba=2ab=d; (a²+b²)²=c²+d²

the final solution follows:

z³=(a²+b²)³=(a²+b²)(c²+d²)=x²+y²

where a, b are any natural numbers and all the rest are calculated as c=a²−b²; d=2ab; x=ac−bd; y=ad+bc (or x=ac+bd; y=ad−bc). Thus, we have established that the original equation z³=x²+y² has an infinite number of solutions in integers and for specific given numbers a, b — two solutions.

It is also clear from this example why one of the Fermat's theorems asserts that:

A prime number in the form 4n+1 and its square can be decomposed into two squares only in one way; its cube and biquadrate only in two; its quadrate-cube and cube-cube only in three etc. to infinity.

Примечания

32

For mathematicians and programmers, the notion of function argument is quite common and has long been generally accepted. In particular, f (x, y, z) denotes a function with variable arguments x, y, z. The definition of the essence of a number through the notion of function arguments makes it very simple, understandable and effective since everything what is known about the number, comes from here and all what this definition does not correspond, should be questioned. This is not just the necessary caution, but also an effective way to test the strength of all kinds of structures, which quietly replace the essence of the number with dubious innovations that make science gormlessly and unsuitable for learning.

33

An exact definition the notion of data does not exist unless it includes a description from the explanatory dictionary. From here follows the uncertainty of its derivative notions such as data format, data processing, data operations etc. Such vague terminology generates a formulaic thinking, indicating that the mind does not develop, but becomes dull and by reaching in this mishmash of empty words critical point, it simply ceases to think. In this work, a definition the notion of “data” is given in Pt. 5.3.2. But for this it is necessary to give the most general definition the notion of information, which in its difficulty will be else greater than the definition the notion of number since the number itself is an information. The advances in this matter are so significant that after they will follow a real technological breakthrough with such potential of efficiency, which will be incomparably higher than which was due to the advent of computers.

34

Computations are not only actions with numbers, but also the application of methods to achieve the final result. Even a machine can cope with actions if the mind equips it with appropriate methods. But if the mind itself becomes like a machine i.e. not aware the methods of calculation, then it is able to create only monsters that will destroy also him selves. Namely to that all is going now because of the complete lack of a solution to the problem of ensuring data security. But the whole problem is that informatics as a science simply does not exist.

35

Specialists who comment on the ancients in their opinion the Euclid's"Elements"and the Diophantus'"Arithmetic", as if spellbound, see but cannot acknowledge the obvious. Neither Euclid nor Diophantus can be the creators the content of these books, this is beyond the power of even modern science. Moreover, these books appeared only in the late Middle Ages when the necessary writing was already developed. The authors of these books were just translators of truly ancient sources belonging to another civilization. Nowadays, people with such abilities are called medium.

36

If from the very beginning we have not decided on the concept of a number and have an idea of it only through prototypes (the number of fingers, or days of the week etc.), then sooner or later we will find that we don’t know anything about numbers and follow the calculations an immense set of empirical methods and rules. However, if initially we have an exact definition the notion of number, then for any calculations, we can use only this definition and the relatively small list of rules following from it. If we ourselves creating the required numbers, we can do this through the function arguments, which are represented in the generally accepted number system. But when it is necessary to calculate unknown numbers corresponding to a given function and task conditions, then special methods will often be required, which without understanding the essence of numbers will be very difficult.

37

The content of Peano’s axioms is as follows: (A1) 1 is a natural number; (A2) For any natural number n there is a natural number denoted by n' and called the number following n; (A3) If m'=n' for any positive integers m, n then m = n; (A4) The number 1 does not follow any natural number i.e. n' is never equal to 1; (A5) If the number 1 has some property P and for any number n with the property P the next number n' also has the property P then any natural number has the property P.

38

In the Euclid's"Elements"there is something similar to this axiom:"1. An unit is that by virtue of each of the things that exist is called one. 2. A number is a multitude composed of units” (Book VII, Definitions).

39

So, count is the nominate starting numbers in a finished (counted) form so that on their basis it becomes possible using a similar method to name any other numbers. All this of course, is not at all difficult, but why is it not taught at school and simply forced to memorize everything without explanation? The answer is very simple — because science simply does not know what a number is, but in any way cannot acknowledge this.

40

The axioms of actions were not separately singled out and are a direct consequence of determining the essence a notion of number. They contribute both to learning and establish a certain responsibility for the validity of any scientific research in the field of numbers. In this sense, the last 6th axiom looks even too categorical. But without this kind of restriction any gibberish can be dragged into the knowledge system and then called it a “breakthrough in science”.

41

The reconstructed proof of Fermat excludes the mistake made by Euclid. However, beginning from Gauss, other well-known proofs the Basic theorem of arithmetic repeat this same mistake. An exception is the proof received by the German mathematician Ernst Zermelo, see Appendix I.

42

Facsimile of the edition with the Cauchy's GTF proof was published by Google under the title MEMIRES DE LA CLASSE DES SCIENCES MATHTÉMATIQUES ET PHYSIQUES DE L’INSTITUT DE France. ANNEES 1813, 1814, 1815: https://books.google.de/books?id=k2pFAAAAcAAJ&pg=PA177#v=onepage&q&f=false What we need is on page 177 under the title DEMONSTRATION DU THÉORÉME GÉNÉRAL DE FERMAT, SUR LES NOMBRES POLYGONES. Par M. A. L. CAUCHY. Lu à l’Académie, le 13 novembre 1815 (see Pics 34, 35). The general proof of Cauchy takes 43 (!!!) pages and this circumstance alone indicates that it does not fit into any textbook. Such creations are not something that students, but also academics are not be available because the first cannot understand anything about them and the second simply do not have the necessary time for this. Then it turns out that such proofs are hardly possible to check how convincing they are i.e. are they any proofs in general? But if Cauchy applied the descent method recommended by Fermat, then the proof would become so convincing that no checks would be required. A very simple conclusion follows from this: The Fermat's Golden Theorem as well as some of his other theorems, still remain unproven.

43

Examples are in many videos from the Internet. However, these examples in no way detract from the merits of professors who know their job perfectly.