He quickly moved on. He found sufficient conditions for the existence of maxima, learned to determine the inflection points, drew tangents to all known curves of the second and third order. A few more years, and he finds a new purely algebraic method for finding quadratures for parabolas and hyperbolas of arbitrary order (that is, integrals of functions of the form y p = Cx q and y p x q \u003d C), calculates areas, volumes, moments of inertia of bodies of revolution. It was a real breakthrough. Feeling this, Fermat begins to seek communication with the mathematical authorities of the time. He is confident and longs for recognition.
In 1636 he wrote the first letter to His Reverend Marin Mersenne: “Holy Father! I am extremely grateful to you for the honor you have done me by giving me the hope that we will be able to talk in writing; ...I will be very glad to hear from you about all the new treatises and books on Mathematics that have appeared in the last five or six years. ...I also found a lot analytical methods for various problems, both numerical and geometric, for which Vieta's analysis is insufficient. All this I will share with you whenever you want, and, moreover, without any arrogance, from which I am freer and more distant than any other person in the world.
Who is Father Mersenne? This is a Franciscan monk, a scientist of modest talents and a wonderful organizer, who for 30 years headed the Parisian mathematical circle, which became the true center of French science. Subsequently, the Mersenne circle, by decree of Louis XIV, will be transformed into the Paris Academy of Sciences. Mersenne tirelessly carried on a huge correspondence, and his cell in the monastery of the Order of the Minims on the Royal Square was a kind of "post office for all the scientists of Europe, from Galileo to Hobbes." Correspondence then replaced scientific journals, which appeared much later. Meetings at Mersenne took place weekly. The core of the circle was made up of the most brilliant natural scientists of that time: Robertville, Pascal Father, Desargues, Midorge, Hardy and, of course, the famous and universally recognized Descartes. Rene du Perron Descartes (Cartesius), a mantle of nobility, two family estates, the founder of Cartesianism, the “father” of analytic geometry, one of the founders of new mathematics, as well as Mersenne’s friend and comrade at the Jesuit College. This wonderful man will be Fermat's nightmare.
Mersenne found Fermat's results interesting enough to bring the provincial into his elite club. The farm immediately strikes up a correspondence with many members of the circle and literally falls asleep with letters from Mersenne himself. In addition, he sends completed manuscripts to the court of pundits: “Introduction to flat and solid places”, and a year later - “Method of finding maxima and minima” and “Answers to questions by B. Cavalieri”. What Fermat expounded was absolutely new, but the sensation did not take place. Contemporaries did not flinch. They didn’t understand much, but they found unambiguous indications that Fermat borrowed the idea of the maximization algorithm from Johannes Kepler’s treatise with the funny title “The New Stereometry of Wine Barrels”. Indeed, in Kepler's reasoning there are phrases like “The volume of the figure is greatest if on both sides of the place the greatest value the decrease is at first insensitive.” But the idea of a small increment of a function near an extremum was not at all in the air. The best analytical minds of that time were not ready for manipulations with small quantities. The fact is that at that time algebra was considered a kind of arithmetic, that is, mathematics of the second grade, a primitive improvised tool developed for the needs of base practice (“only merchants count well”). Tradition prescribed to adhere to purely geometric methods of proofs, dating back to ancient mathematics. Fermat was the first to understand that infinitesimal quantities can be added and reduced, but it is rather difficult to represent them as segments.
It took almost a century for Jean d'Alembert to admit in his famous Encyclopedia: Fermat was the inventor of the new calculus. It is with him that we meet the first application of differentials for finding tangents.” AT late XVIII century, Joseph Louis Comte de Lagrange will express himself even more definitely: “But the geometers - Fermat's contemporaries - did not understand this new kind of calculus. They saw only special cases. And this invention, which appeared shortly before Descartes' Geometry, remained fruitless for forty years. Lagrange is referring to 1674, when Isaac Barrow's "Lectures" were published, covering Fermat's method in detail.
Among other things, it quickly became clear that Fermat was more inclined to formulate new problems than to humbly solve the problems proposed by the meters. In the era of duels, the exchange of tasks between pundits was generally accepted as a form of clarifying issues related to chain of command. However, the Farm clearly does not know the measure. Each of his letters is a challenge containing dozens of complex unsolved problems, and on the most unexpected topics. Here is an example of his style (addressed to Frenicle de Bessy): “Item, what is the smallest square that, when reduced by 109 and added to one, will give a square? If you do not send me the general solution, then send me the quotient for these two numbers, which I chose small so as not to make you very difficult. After I get your answer, I will suggest some other things to you. It is clear, without special reservations, that in my proposal it is required to find whole numbers, because in the case fractional numbers the most insignificant arithmetician could reach the goal.” Fermat often repeated himself, formulating the same questions several times, and openly bluffed, claiming that he had an unusually elegant solution to the proposed problem. There were no direct errors. Some of them were noticed by contemporaries, and some of the insidious statements misled readers for centuries.
Mersenne's circle reacted adequately. Only Robertville, the only member of the circle who had problems with the origin, maintains a friendly tone of letters. The good shepherd Father Mersenne tried to reason with the "Toulouse impudent". But Farm does not intend to make excuses: “Reverend Father! You write to me that the posing of my impossible problems angered and cooled Messrs. Saint-Martin and Frenicle, and that this was the reason for the termination of their letters. However, I want to object to them that what seems impossible at first is actually not, and that there are many problems that, as Archimedes said...” etc.
However, Farm is disingenuous. It was to Frenicle that he sent the problem of finding a right-angled triangle with integer sides whose area is equal to the square of an integer. He sent it, although he knew that the problem obviously had no solution.
The most hostile position towards Fermat was taken by Descartes. In his letter to Mersenne dated 1938 we read: “because I found out that this is the same person who had previously tried to refute my “Dioptric”, and since you informed me that he sent it after he had read my “Geometry ” and in surprise that I did not find the same thing, i.e. (as I have reason to interpret it) sent it with the aim of entering into rivalry and showing that he knows more about it than I do, and since more of your letters, I learned that he had a reputation as a very knowledgeable geometer, then I consider myself obliged to answer him. Descartes will later solemnly designate his answer as “the small trial of Mathematics against Mr. Fermat”.
It is easy to understand what infuriated the eminent scientist. First, in Fermat's reasoning, coordinate axes and the representation of numbers by segments constantly appear - a device that Descartes comprehensively develops in his just published "Geometry". Fermat comes to the idea of replacing the drawing with calculations on his own, in some ways even more consistent than Descartes. Secondly, Fermat brilliantly demonstrates the effectiveness of his method of finding minima on the example of the problem of the shortest path of a light beam, refining and supplementing Descartes with his "Dioptric".
The merits of Descartes as a thinker and innovator are enormous, but let's open the modern “Mathematical Encyclopedia” and look at the list of terms associated with his name: “ Cartesian coordinates” (Leibniz, 1692) , “Cartesian sheet”, “Descartes ovals”. None of his arguments went down in history as Descartes' Theorem. Descartes is primarily an ideologist: he is the founder of a philosophical school, he forms concepts, improves the system letters, but there are few new specific techniques in his creative heritage. In contrast, Pierre Fermat writes little, but on any occasion he can come up with a lot of witty mathematical tricks (see ibid. "Fermat's Theorem", "Fermat's Principle", "Fermat's method of infinite descent"). They probably quite rightly envied each other. The collision was inevitable. With the Jesuit mediation of Mersenne, a war broke out that lasted two years. However, Mersenne turned out to be right before history here too: the fierce battle between the two titans, their tense, to put it mildly, polemic contributed to the understanding of the key concepts of mathematical analysis.
Fermat is the first to lose interest in the discussion. Apparently, he spoke directly with Descartes and never again offended his opponent. In one of his last works, "Synthesis for refraction", the manuscript of which he sent to de la Chaumbra, Fermat mentions "the most learned Descartes" through the word and in every possible way emphasizes his priority in matters of optics. Meanwhile, it was this manuscript that contained the description of the famous "Fermat's principle", which provides an exhaustive explanation of the laws of reflection and refraction of light. Curtseys to Descartes in a work of this level were completely unnecessary.
What happened? Why did Fermat, putting aside pride, went to reconciliation? Reading Fermat's letters of those years (1638 - 1640), one can assume the simplest: during this period, his scientific interests changed drastically. He abandons the fashionable cycloid, ceases to be interested in tangents and areas, and for a long 20 years forgets about his method of finding the maximum. Having great merits in the mathematics of the continuous, Fermat completely immerses himself in the mathematics of the discrete, leaving the hateful geometric drawings to his opponents. Numbers are his new passion. As a matter of fact, the entire "Theory of Numbers", as an independent mathematical discipline, owes its birth entirely to the life and work of Fermat.
<…>After Fermat's death, his son Samuel published in 1670 a copy of Arithmetic belonging to his father under the title "Six books of arithmetic by the Alexandrian Diophantus with comments by L. G. Basche and remarks by P. de Fermat, Senator of Toulouse." The book also included some of Descartes' letters and the full text of Jacques de Bigly's A New Discovery in the Art of Analysis, based on Fermat's letters. The publication was an incredible success. An unprecedented bright world opened up before the astonished specialists. The unexpectedness, and most importantly, the accessibility, democratic nature of Fermat's number-theoretic results gave rise to a lot of imitations. At that time, few people understood how the area of a parabola was calculated, but every student could understand the formulation of Fermat's Last Theorem. A real hunt began for the unknown and lost letters of the scientist. Before late XVII in. Every word of his that was found was published and republished. But the turbulent history of the development of Fermat's ideas was just beginning.
Pierre de Fermat, reading the "Arithmetic" of Diophantus of Alexandria and reflecting on its problems, had the habit of writing down the results of his reflections in the form of brief remarks in the margins of the book. Against the eighth problem of Diophantus in the margins of the book, Fermat wrote: " On the contrary, it is impossible to decompose neither a cube into two cubes, nor a bi-square into two bi-squares, and, in general, no degree greater than a square into two powers with the same exponent. I have discovered a truly marvelous proof of this, but these margins are too narrow for it.» / E.T.Bell "Creators of Mathematics". M., 1979, p.69/. I bring to your attention an elementary proof of the farm theorem, which can be understood by any high school student who is fond of mathematics.
Let us compare Fermat's commentary on the Diophantine problem with the modern formulation of Fermat's great theorem, which has the form of an equation.
« The equation
x n + y n = z n(where n is an integer greater than two)
has no solutions in positive integers»
The comment is in a logical connection with the task, similar to the logical connection of the predicate with the subject. What is affirmed by the problem of Diophantus, on the contrary, is affirmed by Fermat's commentary.
Fermat's comment can be interpreted as follows: if quadratic equation with three unknowns has an infinite number of solutions on the set of all triples of Pythagorean numbers, then, conversely, an equation with three unknowns in a degree greater than the square
There is not even a hint of its connection with the Diophantine problem in the equation. His assertion requires proof, but it does not have a condition from which it follows that it has no solutions in positive integers.
The variants of the proof of the equation known to me are reduced to the following algorithm.
- The equation of Fermat's theorem is taken as its conclusion, the validity of which is verified with the help of proof.
- The same equation is called initial the equation from which its proof must proceed.
The result is a tautology: If an equation has no solutions in positive integers, then it has no solutions in positive integers.". The proof of the tautology is obviously wrong and devoid of any meaning. But it is proved by contradiction.
- An assumption is made that is the opposite of that stated by the equation to be proven. It should not contradict the original equation, but it does. To prove what is accepted without proof, and to accept without proof what is required to be proved, does not make sense.
- Based on the accepted assumption, absolutely correct mathematical operations and actions are performed to prove that it contradicts the original equation and is false.
Therefore, for 370 years now, the proof of the equation of Fermat's Last Theorem has remained an impossible dream of specialists and lovers of mathematics.
I took the equation as the conclusion of the theorem, and the eighth problem of Diophantus and its equation as the condition of the theorem.
"If the equation x 2 + y 2 = z 2
(1) has an infinite set of solutions on the set of all triples of Pythagorean numbers, then, conversely, the equation x n + y n = z n
, where n > 2
(2) has no solutions on the set of positive integers."
Proof.
BUT) Everyone knows that equation (1) has an infinite number of solutions on the set of all triples of Pythagorean numbers. Let us prove that no triple of Pythagorean numbers, which is a solution to equation (1), is a solution to equation (2).
Based on the law of reversibility of equality, the sides of equation (1) are interchanged. Pythagorean numbers (z, x, y) can be interpreted as the lengths of the sides of a right triangle, and the squares (x2, y2, z2) can be interpreted as the areas of squares built on its hypotenuse and legs.
We multiply the squares of equation (1) by an arbitrary height h :
z 2 h = x 2 h + y 2 h (3)
Equation (3) can be interpreted as the equality of the volume of a parallelepiped to the sum of the volumes of two parallelepipeds.
Let the height of three parallelepipeds h = z :
z 3 = x 2 z + y 2 z (4)
The volume of the cube is decomposed into two volumes of two parallelepipeds. We leave the volume of the cube unchanged, and reduce the height of the first parallelepiped to x and the height of the second parallelepiped will be reduced to y . The volume of a cube is greater than the sum of the volumes of two cubes:
z 3 > x 3 + y 3 (5)
On the set of triples of Pythagorean numbers ( x, y, z ) at n=3 there can be no solution to equation (2). Consequently, on the set of all triples of Pythagorean numbers, it is impossible to decompose a cube into two cubes.
Let in equation (3) the height of three parallelepipeds h = z2 :
z 2 z 2 = x 2 z 2 + y 2 z 2 (6)
The volume of a parallelepiped is decomposed into the sum of the volumes of two parallelepipeds.
We leave the left side of equation (6) unchanged. On its right side the height z2
reduce to X
in the first term and up to at 2
in the second term.
Equation (6) turned into the inequality:
The volume of a parallelepiped is decomposed into two volumes of two parallelepipeds.
We leave the left side of equation (8) unchanged.
On the right side of the height zn-2
reduce to xn-2
in the first term and reduce to y n-2
in the second term. Equation (8) turns into the inequality:
| z n > x n + y n | (9) |
On the set of triples of Pythagorean numbers, there cannot be a single solution of equation (2).
Consequently, on the set of all triples of Pythagorean numbers for all n > 2 equation (2) has no solutions.
Obtained "post miraculous proof", but only for triplets Pythagorean numbers. This is lack of evidence and the reason for the refusal of P. Fermat from him.
b) Let us prove that equation (2) has no solutions on the set of triples of non-Pythagorean numbers, which is the family of an arbitrarily taken triple of Pythagorean numbers z=13, x=12, y=5 and the family of an arbitrary triple of positive integers z=21, x=19, y=16
Both triplets of numbers are members of their families:
| (13, 12, 12); (13, 12,11);…; (13, 12, 5) ;…; (13,7, 1);…; (13,1, 1) | (10) | |
| (21, 20, 20); (21, 20, 19);…;(21, 19, 16);…;(21, 1, 1) | (11) |
The number of members of the family (10) and (11) is equal to half the product of 13 by 12 and 21 by 20, i.e. 78 and 210.
Each member of the family (10) contains z = 13 and variables X and at 13 > x > 0 , 13 > y > 0 1
Each member of the family (11) contains z = 21 and variables X and at , which take integer values 21 > x >0 , 21 > y > 0 . The variables decrease sequentially by 1 .
The triples of numbers of the sequence (10) and (11) can be represented as a sequence of inequalities of the third degree:
| 13 3 < 12 3 + 12 3 ;13 3 < 12 3 + 11 3 ;…; 13 3 < 12 3 + 8 3 ; 13 3 > 12 3 + 7 3 ;…; 13 3 > 1 3 + 1 3 | ||
| 21 3 < 20 3 + 20 3 ; 21 3 < 20 3 + 19 3 ; …; 21 3 < 19 3 + 14 3 ; 21 3 > 19 3 + 13 3 ;…; 21 3 > 1 3 + 1 3 |
and in the form of inequalities of the fourth degree:
| 13 4 < 12 4 + 12 4 ;…; 13 4 < 12 4 + 10 4 ; 13 4 > 12 4 + 9 4 ;…; 13 4 > 1 4 + 1 4 | ||
| 21 4 < 20 4 + 20 4 ; 21 4 < 20 4 + 19 4 ; …; 21 4 < 19 4 + 16 4 ;…; 21 4 > 1 4 + 1 4 |
The correctness of each inequality is verified by raising the numbers to the third and fourth powers.
The cube of a larger number cannot be decomposed into two cubes of smaller numbers. It is either less than or greater than the sum of the cubes of the two smaller numbers.
The bi-square of a larger number cannot be decomposed into two bi-squares of smaller numbers. It is either less than or greater than the sum of the bi-squares of smaller numbers.
As the exponent increases, all inequalities, except for the leftmost inequality, have the same meaning:
Inequalities, they all have the same meaning: the degree of the larger number is greater than the sum of the degrees of the smaller two numbers with the same exponent:
| 13n > 12n + 12n ; 13n > 12n + 11n ;…; 13n > 7n + 4n ;…; 13n > 1n + 1n | (12) | |
| 21n > 20n + 20n ; 21n > 20n + 19n ;…; ;…; 21n > 1n + 1n | (13) |
The leftmost term of sequences (12) (13) is the weakest inequality. Its correctness determines the correctness of all subsequent inequalities of the sequence (12) for n > 8 and sequence (13) for n > 14 .
There can be no equality among them. An arbitrary triple of positive integers (21,19,16) is not a solution to equation (2) of Fermat's Last Theorem. If an arbitrary triple of positive integers is not a solution to the equation, then the equation has no solutions on the set of positive integers, which was to be proved.
FROM) Fermat's commentary on the Diophantus problem states that it is impossible to decompose " in general, no power greater than the square, two powers with the same exponent».
Kisses a power greater than a square cannot really be decomposed into two powers with the same exponent. I don't kiss a power greater than the square can be decomposed into two powers with the same exponent.
Any randomly chosen triple of positive integers (z, x, y) may belong to a family, each member of which consists of a constant number z and two numbers less than z . Each member of the family can be represented in the form of an inequality, and all the resulting inequalities can be represented as a sequence of inequalities:
| z n< (z — 1) n + (z — 1) n ; z n < (z — 1) n + (z — 2) n ; …; z n >1n + 1n | (14) |
The sequence of inequalities (14) begins with inequalities for which left-hand side less than the right side, and ends with inequalities whose right side is less than the left side. With increasing exponent n > 2 the number of inequalities on the right side of sequence (14) increases. With an exponent n=k all the inequalities of the left side of the sequence change their meaning and take on the meaning of the inequalities of the right side of the inequalities of the sequence (14). As a result of the increase in the exponent of all inequalities, the left side is greater than the right side:
| z k > (z-1) k + (z-1) k ; z k > (z-1) k + (z-2) k ;…; zk > 2k + 1k ; zk > 1k + 1k | (15) |
With a further increase in the exponent n>k none of the inequalities changes its meaning and does not turn into equality. On this basis, it can be argued that any arbitrarily taken triple of positive integers (z, x, y) at n > 2 , z > x , z > y
In an arbitrary triple of positive integers z can be an arbitrarily large natural number. For all natural numbers, which are no more z , Fermat's Last Theorem is proved.
D) No matter how big the number z , in the natural series of numbers before it there is a large but finite set of integers, and after it there is an infinite set of integers.
Let us prove that the entire infinite set of natural numbers greater than z , form triples of numbers that are not solutions to the equation of Fermat's Last Theorem, for example, an arbitrary triple of positive integers (z+1,x,y) , wherein z + 1 > x and z + 1 > y for all values of the exponent n > 2 is not a solution to the equation of Fermat's Last Theorem.
A randomly chosen triple of positive integers (z + 1, x, y) may belong to a family of triples of numbers, each member of which consists of a constant number z + 1 and two numbers X and at , taking different values, smaller z + 1 . Family members can be represented as inequalities whose constant left side is less than, or greater than, the right side. The inequalities can be arranged in order as a sequence of inequalities:
With a further increase in the exponent n>k to infinity, none of the inequalities in the sequence (17) changes its meaning and does not become an equality. In sequence (16), the inequality formed from an arbitrarily taken triple of positive integers (z + 1, x, y) , can be in its right side in the form (z + 1) n > x n + y n or be on its left side in the form (z+1)n< x n + y n .
In any case, the triple of positive integers (z + 1, x, y) at n > 2 , z + 1 > x , z + 1 > y in sequence (16) is an inequality and cannot be an equality, i.e., it cannot be a solution to the equation of Fermat's Last Theorem.
It is easy and simple to understand the origin of the sequence of power inequalities (16), in which the last inequality of the left side and the first inequality of the right side are inequalities of the opposite sense. On the contrary, it is not easy and difficult for schoolchildren, high school students and high school students to understand how a sequence of inequalities (17) is formed from a sequence of inequalities (16), in which all inequalities have the same meaning.
In sequence (16), increasing the integer degree of inequalities by 1 turns the last inequality on the left side into the first inequality of the opposite meaning on the right side. Thus, the number of inequalities on the ninth side of the sequence decreases, while the number of inequalities on the right side increases. Between the last and first power inequalities of opposite meaning in without fail power equality is found. Its degree cannot be an integer, since there are only non-integer numbers between two consecutive natural numbers. The power equality of a non-integer degree, according to the condition of the theorem, cannot be considered a solution to equation (1).
If in the sequence (16) we continue to increase the degree by 1 unit, then the last inequality of its left side will turn into the first inequality of the opposite meaning of the right side. As a result, there will be no inequalities on the left side and only inequalities on the right side, which will be a sequence of increasing power inequalities (17). Further increase their integer degree by 1 unit only strengthens its power inequalities and categorically excludes the possibility of the appearance of equality in an integer degree.
Therefore, in general, no integer power of a natural number (z+1) of the sequence of power inequalities (17) can be decomposed into two integer powers with the same exponent. Therefore, equation (1) has no solutions on an infinite set of natural numbers, which was to be proved.
Therefore, Fermat's Last Theorem is proved in all generality:
- in section A) for all triplets (z, x, y) Pythagorean numbers (Fermat's discovery is a truly miraculous proof),
- in section C) for all members of the family of any triple (z, x, y) pythagorean numbers,
- in section C) for all triplets of numbers (z, x, y) , not large numbers z
- in section D) for all triples of numbers (z, x, y) natural series of numbers.
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Changes were made on 05.09.2010 |
Which theorems can and which cannot be proven by contradiction
The Explanatory Dictionary of Mathematical Terms defines proof by contradiction of a theorem opposite to the inverse theorem.
“Proof by contradiction is a method of proving a theorem (sentence), which consists in proving not the theorem itself, but its equivalent (equivalent), opposite inverse (reverse to opposite) theorem. Proof by contradiction is used whenever the direct theorem is difficult to prove, but the opposite inverse is easier. When proving by contradiction, the conclusion of the theorem is replaced by its negation, and by reasoning one arrives at the negation of the condition, i.e. to a contradiction, to the opposite (the opposite of what is given; this reduction to absurdity proves the theorem.
Proof by contradiction is very often used in mathematics. The proof by contradiction is based on the law of the excluded middle, which consists in the fact that of the two statements (statements) A and A (negation of A), one of them is true and the other is false./ Explanatory dictionary of mathematical terms: A guide for teachers / O. V. Manturov [and others]; ed. V. A. Ditkina.- M.: Enlightenment, 1965.- 539 p.: ill.-C.112/.
It would not be better to openly declare that the method of proof by contradiction is not a mathematical method, although it is used in mathematics, that it is a logical method and belongs to logic. Is it valid to say that proof by contradiction is "used whenever a direct theorem is difficult to prove", when in fact it is used if, and only if, there is no substitute for it.
The characteristic of the relationship between the direct and inverse theorems also deserves special attention. “An inverse theorem for a given theorem (or to a given theorem) is a theorem in which the condition is the conclusion, and the conclusion is the condition of the given theorem. This theorem in relation to the converse theorem is called the direct theorem (initial). At the same time, the converse theorem to the converse theorem will be the given theorem; therefore, the direct and inverse theorems are called mutually inverse. If the direct (given) theorem is true, then the converse theorem is not always true. For example, if a quadrilateral is a rhombus, then its diagonals are mutually perpendicular (direct theorem). If the diagonals in a quadrilateral are mutually perpendicular, then the quadrilateral is a rhombus - this is not true, i.e., the converse theorem is not true./ Explanatory dictionary of mathematical terms: A guide for teachers / O. V. Manturov [and others]; ed. V. A. Ditkina.- M.: Enlightenment, 1965.- 539 p.: ill.-C.261 /.
This characteristic The relation between direct and inverse theorems does not take into account the fact that the condition of the direct theorem is taken as given, without proof, so that its correctness is not guaranteed. The condition of the inverse theorem is not taken as given, since it is the conclusion of the proven direct theorem. Its correctness is confirmed by the proof of the direct theorem. This essential logical difference between the conditions of the direct and inverse theorems turns out to be decisive in the question of which theorems can and which cannot be proved by the logical method from the contrary.
Let's assume that there is a direct theorem in mind, which can be proved by the usual mathematical method, but it is difficult. Let's formulate it in general view in short form So: from BUT should E . Symbol BUT has the value of the given condition of the theorem, accepted without proof. Symbol E is the conclusion of the theorem to be proved.
We will prove the direct theorem by contradiction, logical method. The logical method proves a theorem that has not mathematical condition, and logical condition. It can be obtained if mathematical condition theorems from BUT should E , supplement with the opposite condition from BUT it does not follow E .
As a result, a logical contradictory condition of the new theorem was obtained, which includes two parts: from BUT should E and from BUT it does not follow E . The resulting condition of the new theorem corresponds to the logical law of the excluded middle and corresponds to the proof of the theorem by contradiction.
According to the law, one part of the contradictory condition is false, another part is true, and the third is excluded. The proof by contradiction has its own task and goal to establish exactly which part of the two parts of the condition of the theorem is false. As soon as the false part of the condition is determined, it will be established that the other part is the true part, and the third is excluded.
According to explanatory dictionary mathematical terms “proof is reasoning, during which the truth or falsity of any statement (judgment, statement, theorem) is established”. Proof contrary there is a discussion in the course of which it is established falsity(absurdity) of the conclusion that follows from false conditions of the theorem being proved.
Given: from BUT should E and from BUT it does not follow E .
Prove: from BUT should E .
Proof: The logical condition of the theorem contains a contradiction that requires its resolution. The contradiction of the condition must find its resolution in the proof and its result. The result turns out to be false if the reasoning is flawless and infallible. The reason for a false conclusion with logically correct reasoning can only be a contradictory condition: from BUT should E and from BUT it does not follow E .
There is no shadow of a doubt that one part of the condition is false, and the other in this case is true. Both parts of the condition have the same origin, are accepted as given, assumed, equally possible, equally permissible, etc. In the course of logical reasoning, not a single logical feature, which would distinguish one part of the condition from another. Therefore, to the same extent, from BUT should E and maybe from BUT it does not follow E . Statement from BUT should E may be false, then the statement from BUT it does not follow E will be true. Statement from BUT it does not follow E may be false, then the statement from BUT should E will be true.
Therefore, it is impossible to prove the direct theorem by contradiction method.
Now we will prove the same direct theorem by the usual mathematical method.
Given: BUT .
Prove: from BUT should E .
Proof.
1. From BUT should B
2. From B should AT (according to the previously proved theorem)).
3. From AT should G (according to the previously proved theorem).
4. From G should D (according to the previously proved theorem).
5. From D should E (according to the previously proved theorem).
Based on the law of transitivity, from BUT should E . The direct theorem is proved by the usual method.
Let the proven direct theorem have a correct converse theorem: from E should BUT .
Let's prove it by ordinary mathematical method. The proof of the inverse theorem can be expressed in symbolic form as an algorithm of mathematical operations.
Given: E
Prove: from E should BUT .
Proof.
1. From E should D
2. From D should G (by the previously proved inverse theorem).
3. From G should AT (by the previously proved inverse theorem).
4. From AT it does not follow B (the converse is not true). That's why from B it does not follow BUT .
In this situation, it makes no sense to continue the mathematical proof of the inverse theorem. The reason for the situation is logical. It is impossible to replace an incorrect inverse theorem with anything. Therefore, this inverse theorem cannot be proved by the usual mathematical method. All hope is to prove this inverse theorem by contradiction.
In order to prove it by contradiction, it is required to replace its mathematical condition with a logical contradictory condition, which in its meaning contains two parts - false and true.
Inverse theorem claims: from E it does not follow BUT . Her condition E , from which follows the conclusion BUT , is the result of proving the direct theorem by the usual mathematical method. This condition must be retained and supplemented with the statement from E should BUT . As a result of the addition, a contradictory condition of the new inverse theorem is obtained: from E should BUT and from E it does not follow BUT . Based on this logically contradictory condition, the converse theorem can be proved by the correct logical reasoning only, and only, logical opposite method. In a proof by contradiction, any mathematical actions and operations are subordinate to logical ones and therefore do not count.
In the first part of the contradictory statement from E should BUT condition E was proved by the proof of the direct theorem. In the second part from E it does not follow BUT condition E was assumed and accepted without proof. One of them is false and the other is true. It is required to prove which of them is false.
We prove with the correct logical reasoning and find that its result is a false, absurd conclusion. The reason for a false logical conclusion is the contradictory logical condition of the theorem, which contains two parts - false and true. The false part can only be a statement from E it does not follow BUT , wherein E accepted without proof. This is what distinguishes it from E statements from E should BUT , which is proved by the proof of the direct theorem.
Therefore, the statement is true: from E should BUT , which was to be proved.
Conclusion: only that converse theorem is proved by the logical method from the contrary, which has a direct theorem proved by the mathematical method and which cannot be proved by the mathematical method.
The conclusion obtained acquires an exceptional importance in relation to the method of proof by contradiction of Fermat's great theorem. The overwhelming majority of attempts to prove it are based not on the usual mathematical method, but on the logical method of proving by contradiction. The proof of Fermat Wiles' Great Theorem is no exception.
Dmitry Abrarov in his article "Fermat's Theorem: the Phenomenon of Wiles' Proofs" published a commentary on the proof of Fermat's Last Theorem by Wiles. According to Abrarov, Wiles proves Fermat's Last Theorem with the help of a remarkable finding by the German mathematician Gerhard Frey (b. 1944) relating a potential solution to Fermat's equation x n + y n = z n
, where n > 2
, with another completely different equation. This new equation is given by a special curve (called the Frey elliptic curve). The Frey curve is given by a very simple equation:
.
“It was precisely Frey who compared to every solution (a, b, c) Fermat's equation, that is, numbers satisfying the relation a n + b n = c n the above curve. In this case, Fermat's Last Theorem would follow."(Quote from: Abrarov D. "Fermat's Theorem: the phenomenon of Wiles proof")
In other words, Gerhard Frey suggested that the equation of Fermat's Last Theorem x n + y n = z n
, where n > 2
, has solutions in positive integers. The same solutions are, by Frey's assumption, the solutions of his equation
y 2 + x (x - a n) (y + b n) = 0
, which is given by its elliptic curve.
Andrew Wiles accepted this remarkable discovery of Frey and, with its help, through mathematical method proved that this finding, that is, Frey's elliptic curve, does not exist. Therefore, there is no equation and its solutions that are given by a non-existent elliptic curve. Therefore, Wiles should have concluded that there is no equation of Fermat's Last Theorem and Fermat's Theorem itself. However, he takes the more modest conclusion that the equation of Fermat's Last Theorem has no solutions in positive integers.
It may be an undeniable fact that Wiles accepted an assumption that is directly opposite in meaning to what is stated by Fermat's Last Theorem. It obliges Wiles to prove Fermat's Last Theorem by contradiction. Let's follow his example and see what happens from this example.
Fermat's Last Theorem states that the equation x n + y n = z n , where n > 2 , has no solutions in positive integers.
According to the logical method of proof by contradiction, this statement is preserved, accepted as given without proof, and then supplemented with a statement opposite in meaning: the equation x n + y n = z n , where n > 2 , has solutions in positive integers.
The hypothesized statement is also accepted as given, without proof. Both statements, considered from the point of view of the basic laws of logic, are equally admissible, equal in rights and equally possible. By correct reasoning, it is required to establish which of them is false, in order to then establish that the other statement is true.
Correct reasoning ends with a false, absurd conclusion, the logical cause of which can only be a contradictory condition of the theorem being proved, which contains two parts of a directly opposite meaning. They were the logical cause of the absurd conclusion, the result of proof by contradiction.
However, in the course of logically correct reasoning, not a single sign was found by which it would be possible to establish which particular statement is false. It can be a statement: the equation x n + y n = z n , where n > 2 , has solutions in positive integers. On the same basis, it can be the statement: the equation x n + y n = z n , where n > 2 , has no solutions in positive integers.
As a result of the reasoning, there can be only one conclusion: Fermat's Last Theorem cannot be proven by contradiction.
It would be a very different matter if Fermat's Last Theorem were an inverse theorem that has a direct theorem proved by the usual mathematical method. In this case, it could be proven by contradiction. And since it is a direct theorem, its proof must be based not on the logical method of proof by contradiction, but on the usual mathematical method.
According to D. Abrarov, the most famous of modern Russian mathematicians Academician V. I. Arnold reacted to Wiles's proof "actively skeptical". The academician stated: “this is not real mathematics - real mathematics is geometric and has strong links with physics.”
By contradiction, it is impossible to prove either that the equation of Fermat's Last Theorem has no solutions, or that it has solutions. Wiles' mistake is not mathematical, but logical - the use of proof by contradiction where its use does not make sense and does not prove Fermat's Last Theorem.
Fermat's Last Theorem is not proved with the help of the usual mathematical method, if it is given: the equation x n + y n = z n , where n > 2 , has no solutions in positive integers, and if it is required to prove in it: the equation x n + y n = z n , where n > 2 , has no solutions in positive integers. In this form, there is not a theorem, but a tautology devoid of meaning.
Note. My BTF proof was discussed on one of the forums. One of the participants in Trotil, a specialist in number theory, made the following authoritative statement entitled: "A brief retelling of what Mirgorodsky did." I quote it verbatim:
« BUT. He proved that if z 2 \u003d x 2 + y , then z n > x n + y n . This is a well-known and quite obvious fact.
AT. He took two triples - Pythagorean and non-Pythagorean and showed by simple enumeration that for a specific, specific family of triples (78 and 210 pieces) BTF is performed (and only for it).
FROM. And then the author omitted the fact that from < in a subsequent degree may be = , not only > . A simple counterexample is the transition n=1 in n=2 in a Pythagorean triple.
D. This point does not contribute anything essential to the BTF proof. Conclusion: BTF has not been proven.”
I will consider his conclusion point by point.
BUT. In it, the BTF is proved for the entire infinite set of triples of Pythagorean numbers. Proven by a geometric method, which, as I believe, was not discovered by me, but rediscovered. And it was opened, as I believe, by P. Fermat himself. Fermat might have had this in mind when he wrote:
"I have discovered a truly marvelous proof of this, but these margins are too narrow for it." This assumption of mine is based on the fact that in the Diophantine problem, against which, in the margins of the book, Fermat wrote, we are talking about solutions to the Diophantine equation, which are triples of Pythagorean numbers.
An infinite set of triples of Pythagorean numbers are solutions to the Diophatian equation, and in Fermat's theorem, on the contrary, none of the solutions can be a solution to the equation of Fermat's theorem. And Fermat's truly miraculous proof has a direct bearing on this fact. Later, Fermat could extend his theorem to the set of all natural numbers. On the set of all natural numbers, BTF does not belong to the "set of exceptionally beautiful theorems". This is my assumption, which can neither be proved nor disproved. It can be both accepted and rejected.
AT. In this paragraph, I prove that both the family of an arbitrarily taken Pythagorean triple of numbers and the family of an arbitrarily taken non-Pythagorean triple of numbers BTF is satisfied. This is a necessary, but insufficient and intermediate link in my proof of the BTF. The examples I have taken of the family of a triple of Pythagorean numbers and the family of a triple of non-Pythagorean numbers have the meaning of specific examples that presuppose and do not exclude the existence of similar other examples.
Trotil's statement that I "showed by simple enumeration that for a specific, specific family of triples (78 and 210 pieces) BTF is fulfilled (and only for it) is without foundation. He cannot refute the fact that I could just as well take other examples of Pythagorean and non-Pythagorean triples to get a specific family of one and the other triple.
Whatever pair of triples I take, checking their suitability for solving the problem can be carried out, in my opinion, only by the method of "simple enumeration". Any other method is not known to me and is not required. If he did not like Trotil, then he should have suggested another method, which he does not. Without offering anything in return, to condemn the "simple search", which in this case irreplaceable, incorrect.
FROM. I omitted = between< и < на основании того, что в доказательстве БТФ рассматривается уравнение z 2 \u003d x 2 + y (1), in which the degree n > 2 — whole positive number. From the equality between the inequalities it follows obligatory consideration of equation (1) with a non-integer value of the degree n > 2 . Trotil counting compulsory consideration of equality between inequalities, actually considers necessary in the BTF proof, consideration of equation (1) with non-integer degree value n > 2 . I did this for myself and found that equation (1) with non-integer degree value n > 2 has a solution of three numbers: z, (z-1), (z-1) with a non-integer exponent.
There are not many people in the world who have never heard of Fermat's Last Theorem - perhaps this is the only mathematical problem, which received such wide popularity and became a real legend. It is mentioned in many books and films, while the main context of almost all mentions is the impossibility of proving the theorem.
Yes, this theorem is very famous and in a sense has become an “idol” worshiped by amateur and professional mathematicians, but few people know that its proof was found, and this happened back in 1995. But first things first.
So, Fermat's Last Theorem (often called Fermat's last theorem), formulated in 1637 by the brilliant French mathematician Pierre Fermat, is very simple in nature and understandable to any person with a secondary education. It says that the formula a to the power of n + b to the power of n \u003d c to the power of n has no natural (that is, non-fractional) solutions for n> 2. Everything seems to be simple and clear, but the best mathematicians and ordinary amateurs fought over searching for a solution for more than three and a half centuries.
Why is she so famous? Now let's find out...
Are there few proven, unproved, and yet unproven theorems? The thing is that Fermat's Last Theorem is the biggest contrast between the simplicity of the formulation and the complexity of the proof. Fermat's Last Theorem is an incredibly difficult task, and yet its formulation can be understood by everyone with 5th grade high school, but the proof is not even any professional mathematician. Neither in physics, nor in chemistry, nor in biology, nor in the same mathematics is there a single problem that would be formulated so simply, but remained unresolved for so long. 2. What does it consist of?
Let's start with Pythagorean pants The wording is really simple - at first glance. As we know from childhood, Pythagorean pants all sides are equal." The problem looks so simple because it was based on a mathematical statement that everyone knows - the Pythagorean theorem: in any right triangle a square built on the hypotenuse, is equal to the sum squares built on legs.
In the 5th century BC. Pythagoras founded the Pythagorean brotherhood. The Pythagoreans, among other things, studied integer triples satisfying the equation x²+y²=z². They proved that there are infinitely many Pythagorean triples and got general formulas to find them. They must have tried looking for threes or more. high degrees. Convinced that this did not work, the Pythagoreans abandoned their futile attempts. The members of the fraternity were more philosophers and aesthetes than mathematicians.
That is, it is easy to pick up a set of numbers that perfectly satisfy the equality x² + y² = z²
Starting from 3, 4, 5 - indeed, the elementary school student understands that 9 + 16 = 25.
Or 5, 12, 13: 25 + 144 = 169. Great.
Well, it turns out they don't. This is where the trick starts. Simplicity is apparent, because it is difficult to prove not the presence of something, but, on the contrary, the absence. When it is necessary to prove that there is a solution, one can and should simply present this solution.
It is more difficult to prove the absence: for example, someone says: such and such an equation has no solutions. Put him in a puddle? easy: bam - and here it is, the solution! (give a solution). And that's it, the opponent is defeated. How to prove absence?
To say: "I did not find such solutions"? Or maybe you didn't search well? And what if they are, only very large, well, such that even a super-powerful computer does not yet have enough strength? This is what is difficult.
In a visual form, this can be shown as follows: if we take two squares of suitable sizes and disassemble them into unit squares, then a third square is obtained from this bunch of unit squares (Fig. 2): 
And let's do the same with the third dimension (Fig. 3) - it doesn't work. There are not enough cubes, or extra ones remain:
But the mathematician of the 17th century, the Frenchman Pierre de Fermat, enthusiastically explored general equation x n + y n \u003d z n. And, finally, he concluded: for n>2 integer solutions do not exist. Fermat's proof is irretrievably lost. Manuscripts are on fire! All that remains is his remark in Diophantus' Arithmetic: "I have found a truly amazing proof of this proposition, but the margins here are too narrow to contain it."
Actually, a theorem without proof is called a hypothesis. But Fermat has a reputation for never being wrong. Even if he did not leave proof of any statement, it was subsequently confirmed. In addition, Fermat proved his thesis for n=4. So the hypothesis of the French mathematician went down in history as Fermat's Last Theorem.
After Fermat, such great minds as Leonhard Euler worked on the search for proof (in 1770 he proposed a solution for n = 3),
Adrien Legendre and Johann Dirichlet (these scientists jointly found a proof for n = 5 in 1825), Gabriel Lame (who found a proof for n = 7) and many others. By the mid-1980s, it became clear that academia is on the way to finally solving Fermat's Last Theorem, but it was not until 1993 that mathematicians saw and believed that the three-century saga of finding a proof of Fermat's Last Theorem was almost over.
It is easy to show that it suffices to prove Fermat's theorem only for prime n: 3, 5, 7, 11, 13, 17, … For composite n, the proof remains valid. But also prime numbers infinitely many...
In 1825, using the method of Sophie Germain, the women mathematicians, Dirichlet and Legendre independently proved the theorem for n=5. In 1839, the Frenchman Gabriel Lame showed the truth of the theorem for n=7 using the same method. Gradually, the theorem was proved for almost all n less than a hundred.
Finally, the German mathematician Ernst Kummer showed in a brilliant study that the methods of mathematics in the 19th century cannot prove the theorem in general form. The prize of the French Academy of Sciences, established in 1847 for the proof of Fermat's theorem, remained unassigned.
In 1907, the wealthy German industrialist Paul Wolfskel decided to take his own life because of unrequited love. Like a true German, he set the date and time of the suicide: exactly at midnight. On the last day, he made a will and wrote letters to friends and relatives. Business ended before midnight. I must say that Paul was interested in mathematics. Having nothing to do, he went to the library and began to read Kummer's famous article. It suddenly seemed to him that Kummer had made a mistake in his reasoning. Wolfskehl, with a pencil in his hand, began to analyze this part of the article. Midnight passed, morning came. The gap in the proof was filled. And the very reason for suicide now looked completely ridiculous. Paul tore up the farewell letters and rewrote the will.
He soon died of natural causes. The heirs were pretty surprised: 100,000 marks (more than 1,000,000 current pounds sterling) were transferred to the account of the Royal scientific society Göttingen, which in the same year announced a competition for the Wolfskel Prize. 100,000 marks relied on the prover of Fermat's theorem. Not a pfennig was supposed to be paid for the refutation of the theorem ...
Most professional mathematicians considered the search for a proof of Fermat's Last Theorem to be a lost cause and resolutely refused to waste time on such a futile exercise. But amateurs frolic to glory. A few weeks after the announcement, an avalanche of "evidence" hit the University of Göttingen. Professor E. M. Landau, whose duty was to analyze the evidence sent, distributed cards to his students:
Dear (s). . . . . . . .
Thank you for the manuscript you sent with the proof of Fermat's Last Theorem. The first error is on page ... at line ... . Because of it, the whole proof loses its validity.
Professor E. M. Landau
In 1963, Paul Cohen, drawing on Gödel's findings, proved the unsolvability of one of Hilbert's twenty-three problems, the continuum hypothesis. What if Fermat's Last Theorem is also unsolvable?! But the true fanatics of the Great Theorem did not disappoint at all. The advent of computers unexpectedly gave mathematicians a new method of proof. After World War II, groups of programmers and mathematicians proved Fermat's Last Theorem for all values of n up to 500, then up to 1,000, and later up to 10,000.
In the 80s, Samuel Wagstaff raised the limit to 25,000, and in the 90s, mathematicians claimed that Fermat's Last Theorem was true for all values of n up to 4 million. But if even a trillion trillion is subtracted from infinity, it does not become smaller. Mathematicians are not convinced by statistics. Proving the Great Theorem meant proving it for ALL n going to infinity.
In 1954, two young Japanese mathematician friends took up the study of modular forms. These forms generate series of numbers, each - its own series. By chance, Taniyama compared these series with series generated by elliptic equations. They matched! But modular forms are geometric objects, while elliptic equations are algebraic. Between such different objects never found a connection.
Nevertheless, after careful testing, friends put forward a hypothesis: every elliptic equation has a twin - a modular form, and vice versa. It was this hypothesis that became the foundation of a whole trend in mathematics, but until the Taniyama-Shimura hypothesis was proven, the whole building could collapse at any moment.
In 1984, Gerhard Frey showed that a solution to Fermat's equation, if it exists, can be included in some elliptic equation. Two years later, Professor Ken Ribet proved that this hypothetical equation cannot have a counterpart in the modular world. Henceforth, Fermat's Last Theorem was inextricably linked with the Taniyama-Shimura hypothesis. Having proved that any elliptic curve is modular, we conclude that there is no elliptic equation with a solution to Fermat's equation, and Fermat's Last Theorem would be immediately proved. But for thirty years, it was not possible to prove the Taniyama-Shimura hypothesis, and there were less and less hopes for success.
In 1963, when he was only ten years old, Andrew Wiles was already fascinated by mathematics. When he learned about the Great Theorem, he realized that he could not deviate from it. As a schoolboy, student, graduate student, he prepared himself for this task.
Upon learning of Ken Ribet's findings, Wiles threw himself into proving the Taniyama-Shimura conjecture. He decided to work in complete isolation and secrecy. “I understood that everything that has something to do with Fermat’s Last Theorem is of too much interest ... Too many viewers deliberately interfere with the achievement of the goal.” Seven years of hard work paid off, Wiles finally completed the proof of the Taniyama-Shimura conjecture.
In 1993, the English mathematician Andrew Wiles presented to the world his proof of Fermat's Last Theorem (Wiles read his sensational report at a conference at the Sir Isaac Newton Institute in Cambridge.), work on which lasted more than seven years.
While the hype continued in the press, serious work began to verify the evidence. Each piece of evidence must be carefully examined before the proof can be considered rigorous and accurate. Wiles spent a hectic summer waiting for reviewers' feedback, hoping he could win their approval. At the end of August, experts found an insufficiently substantiated judgment.
It turned out that this decision contains a gross error, although in general it is true. Wiles did not give up, called on the help of a well-known specialist in number theory Richard Taylor, and already in 1994 they published a corrected and supplemented proof of the theorem. The most amazing thing is that this work took up as many as 130 (!) pages in the Annals of Mathematics mathematical journal. But the story did not end there either - the last point was made only in the following year, 1995, when the final and “ideal”, from a mathematical point of view, version of the proof was published.
“...half a minute after the start of the festive dinner on the occasion of her birthday, I gave Nadia the manuscript of the complete proof” (Andrew Wales). Did I mention that mathematicians are strange people?
This time there was no doubt about the proof. Two articles were subjected to the most careful analysis and in May 1995 were published in the Annals of Mathematics.
A lot of time has passed since that moment, but there is still an opinion in society about the unsolvability of Fermat's Last Theorem. But even those who know about the proof found continue to work in this direction - few people are satisfied that the Great Theorem requires a solution of 130 pages!
Therefore, now the forces of so many mathematicians (mostly amateurs, not professional scientists) are thrown in search of a simple and concise proof, but this path, most likely, will not lead anywhere ...
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