Lecture 6. Application of derivatives to the study of functions
If the function f(x) has a derivative at each point of the segment [ A, b], then its behavior can be studied using the derivative f"(X).
Let's look at the basic theorems of differential calculus that underlie derivative applications.
Fermat's theorem
Theorem(Farm) ( about the equality to zero of the derivative ). If function f(x), differentiable on the interval (a, b) and reaches its largest or smallest value at point c є ( a, b), then the derivative of the function at this point is zero, i.e. f"(With) = 0.
Proof. Let the function f(x) is differentiable on the interval ( a, b) and at the point X = With takes the greatest value M at With є ( a, b) (Fig. 1), i.e.
f(With) ≥ f(x) or f(x) – f(c) ≤ 0 or f(s +Δ X) – f(With) ≤ 0.
Derivative f"(x) at point X = With: .
If x> c, Δ X> 0 (i.e. Δ X→ 0 to the right of the point With), That 
and therefore f"(With) ≤ 0.
If x< с
, Δ X< 0 (т.е. ΔX→ 0 to the left of the point With), That 
, from which it follows that f"(With) ≥ 0.
By condition f(x) is differentiable at the point With, therefore, its limit at x→ With does not depend on the choice of direction of approach of the argument x to the point With, i.e. .
We obtain a system from which it follows f"(With) = 0.
In case f(With) = T(those. f(x) takes at point With smallest value), the proof is similar. The theorem has been proven.
Geometric meaning of Fermat's theorem: at the point of the largest or smallest value achieved within the interval, the tangent to the graph of the function is parallel to the x-axis.
File FERMA-KDVar © N. M. Koziy, 2008
Certificate of Ukraine No. 27312
BRIEF PROOF OF FERmat's Last Theorem
Fermat's Last Theorem is formulated as follows: Diophantine equation (http://soluvel.okis.ru/evrika.html):
A n + B n = C n * /1/
Where n- a positive integer greater than two has no solution in positive integers A , B , WITH .
PROOF
From the formulation of Fermat’s Last Theorem it follows: if n is a positive integer greater than two, then provided that two of the three numbers A , IN or WITH- positive integers, one of these numbers is not a positive integer.
We construct the proof based on the fundamental theorem of arithmetic, which is called the “unique factorization theorem” or “the uniqueness theorem of factorization of composite integers.” Odd and even exponents are possible n . Let's consider both cases.
1. Case one: exponent n - odd number.
In this case, the expression /1/ is transformed according to known formulas as follows:
A n + IN n = WITH n /2/
We believe that A And B– positive integers.
Numbers A , IN And WITH must be mutually prime numbers.
From equation /2/ it follows that for given values of numbers A And B factor ( A + B ) n , WITH.
Let's assume that the number WITH - positive integer. Taking into account the accepted conditions and the fundamental theorem of arithmetic, the condition must be satisfied :
WITH n = A n + B n =(A+B) n ∙ D n , / 3/
where is the factor Dn D
From equation /3/ it follows:
From equation /3/ it also follows that the number [ Cn = A n + Bn ] provided that the number WITH ( A + B ) n. However, it is known that:
A n + Bn < ( A + B ) n /5/
Hence:
- a fractional number less than one. /6/
A fractional number.
n
For odd exponents n >2 number:
< 1- дробное число, не являющееся рациональной дробью.
From the analysis of equation /2/ it follows that for an odd exponent n number:
WITH n = A n + IN n = (A+B)
consists of two specific algebraic factors, and for any value of the exponent n the algebraic factor remains unchanged ( A + B ).
Thus, Fermat's Last Theorem has no solution in positive integers for odd exponents n >2.
2. Case two: exponent n - even number .
The essence of Fermat's last theorem will not change if we rewrite equation /1/ as follows:
A n = Cn - Bn /7/
In this case, equation /7/ is transformed as follows:
A n = C n - B n = ( WITH +B)∙(C n-1 + C n-2 · B+ C n-3 ∙ B 2 +…+ C ∙ Bn -2 + Bn -1 ). /8/
We accept that WITH And IN- whole numbers.
From equation /8/ it follows that for given values of numbers B And C factor (C+ B ) has the same value for any value of the exponent n , therefore it is a divisor of the number A .
Let's assume that the number A– an integer. Taking into account the accepted conditions and the fundamental theorem of arithmetic, the condition must be satisfied :
A n = C n - Bn =(C+ B ) n ∙ Dn , / 9/
where is the factor Dn must be an integer and therefore the number D must also be an integer.
From equation /9/ it follows:
/10/
From equation /9/ it also follows that the number [ A n = WITH n - Bn ] provided that the number A– an integer, must be divisible by a number (C+ B ) n. However, it is known that:
WITH n - Bn < (С+ B ) n /11/
Hence:
- a fractional number less than one. /12/
A fractional number.
It follows that for an odd value of the exponent n equation /1/ of Fermat's last theorem has no solution in positive integers.
For even exponents n >2 number:
< 1- дробное число, не являющееся рациональной дробью.
Thus, Fermat's last theorem has no solution in positive integers and for even exponents n >2.
The general conclusion follows from the above: equation /1/ of Fermat’s last theorem has no solution in positive integers A, B And WITH provided that the exponent n >2.
ADDITIONAL RATIONALE
In the case where the exponent n – even number, algebraic expression ( Cn - Bn ) decomposes into algebraic factors:
C 2 – B 2 =(C-B) ∙ (C+B); /13/
C 4 – B 4 = ( C-B) ∙ (C+B) (C 2 + B 2);/14/
C 6 – B 6 =(C-B) ∙ (C+B) · (C 2 –CB + B 2) ∙ (C 2 +CB+ B 2) ; /15/
C 8 – B 8= (C-B) ∙ (C+B) ∙ (C 2 + B 2) ∙ (C 4 + B 4)./16/
Let's give examples in numbers.
EXAMPLE 1: B=11; C=35.
C 2 – B 2 = (2 2 ∙ 3) ∙ (2 23) = 2 4 3 23;
C 4 – B 4 = (2 2 ∙ 3) ∙ (2 23) (2 673) = 2 4 3 23 673;
C 6 – B 6 = (2 2 ∙ 3) ∙ (2 · 23) · (31 2) · (3 · 577) =2 ∙ 3 ∙ 23 ∙ 31 2 ∙ 577;
C 8 – B 8 = (2 2 ∙ 3) ∙ (2 23) (2 673) ∙ (2 75633) = 2 5 ∙ 3 ∙ 23 ∙673 ∙ 75633 .
EXAMPLE 2: B=16; C=25.
C 2 – B 2 = (3 2) ∙ (41) = 3 2 ∙ 41;
C 4 – B 4 = (3 2) ∙ (41) · (881) =3 2 ∙ 41 · 881;
C 6 – B 6 = (3 2) ∙ (41) ∙ (2 2 ∙ 3) ∙ (13 37) (3 ∙ 7 61) = 3 3 7 ∙ 13 37 ∙ 41 ∙ 61;
C 8 – B 8 = (3 2) ∙ (41) ∙ (881) ∙ (17 26833) = 3 2 ∙ 41 ∙ 881 ∙ 17 26833.
From the analysis of equations /13/, /14/, /15/ and /16/ and the corresponding numerical examples it follows:
For a given exponent n , if it is an even number, the number A n = C n - Bn decomposes into a well-defined number of well-defined algebraic factors;
For any exponent n , if it is an even number, in the algebraic expression ( Cn - Bn ) there are always multipliers ( C - B ) And ( C + B ) ;
Each algebraic factor corresponds to a completely definite numerical factor;
For given numbers IN And WITH numeric factors can be prime numbers or composite numeric factors;
Each composite numeric factor is a product of prime numbers that are partially or completely absent from other composite numeric factors;
The size of prime numbers in the composition of composite numerical factors increases with the increase of these factors;
The largest composite numerical factor corresponding to the largest algebraic factor includes the largest prime number to a power less than the exponent n(most often in the first degree).
CONCLUSIONS: Additional evidence supports the conclusion that Fermat's Last Theorem has no solution in positive integers.
mechanical engineer
Judging by the popularity of the query "Fermat's theorem - short proof" this mathematical problem really interests many people. This theorem was first stated by Pierre de Fermat in 1637 on the edge of a copy of Arithmetic, where he claimed that he had a solution that was too large to fit on the edge.
The first successful proof was published in 1995, a complete proof of Fermat's theorem by Andrew Wiles. It was described as "stunning progress" and led Wiles to receive the Abel Prize in 2016. While described relatively briefly, the proof of Fermat's theorem also proved much of the modularity theorem and opened up new approaches to numerous other problems and effective methods for raising modularity. These achievements advanced mathematics by 100 years. The proof of Fermat's little theorem is not something out of the ordinary today.
The unsolved problem stimulated the development of algebraic number theory in the 19th century and the search for a proof of the modularity theorem in the 20th century. It is one of the most notable theorems in the history of mathematics and, prior to the complete proof of Fermat's last theorem by division, it was in the Guinness Book of Records as the "hardest mathematical problem", one of the features of which is that it has the largest number of failed proofs.
Historical reference
The Pythagorean equation x 2 + y 2 = z 2 has an infinite number of positive integer solutions for x, y and z. These solutions are known as Pythagorean trinities. Around 1637, Fermat wrote on the margin of a book that the more general equation a n + b n = c n had no solutions in natural numbers if n was an integer greater than 2. Although Fermat himself claimed to have a solution to his problem, he did not leave no details about her proof. The elementary proof of Fermat's theorem, stated by its creator, was rather his boastful invention. The book of the great French mathematician was discovered 30 years after his death. This equation, called Fermat's Last Theorem, remained unsolved in mathematics for three and a half centuries.
The theorem eventually became one of the most notable unsolved problems in mathematics. Attempts to prove this sparked significant developments in number theory, and over time, Fermat's Last Theorem became known as an unsolved problem in mathematics.
Brief history of evidence
If n = 4, as Fermat himself proved, it is enough to prove the theorem for indices n, which are prime numbers. Over the next two centuries (1637-1839) the conjecture was proven only for the prime numbers 3, 5 and 7, although Sophie Germain updated and proved an approach that applied to the entire class of prime numbers. In the mid-19th century, Ernst Kummer expanded on this and proved the theorem for all regular primes, causing irregular primes to be analyzed individually. Building on Kummer's work and using sophisticated computer research, other mathematicians were able to expand the solution to the theorem, aiming to cover all major exponents up to four million, but the proof for all exponents was still unavailable (meaning that mathematicians generally considered the solution to the theorem impossible, extremely difficult, or unattainable with current knowledge).
Work by Shimura and Taniyama
In 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected that there was a connection between elliptic curves and modular forms, two completely different areas of mathematics. Known at the time as the Taniyama-Shimura-Weil conjecture and (eventually) as the modularity theorem, it stood on its own, with no apparent connection to Fermat's last theorem. It was widely regarded as an important mathematical theorem in its own right, but was considered (like Fermat's theorem) impossible to prove. At the same time, the proof of Fermat's great theorem (by the method of division and the use of complex mathematical formulas) was carried out only half a century later.
In 1984, Gerhard Frey noticed an obvious connection between these two previously unrelated and unresolved problems. Complete proof that the two theorems were closely related was published in 1986 by Ken Ribet, who built on a partial proof by Jean-Pierre Serres, who proved all but one part, known as the "epsilon conjecture". Simply put, these works by Frey, Serres and Ribe showed that if the modularity theorem could be proven for at least a semistable class of elliptic curves, then the proof of Fermat's last theorem would also be discovered sooner or later. Any solution that can contradict Fermat's last theorem can also be used to contradict the modularity theorem. Therefore, if the modularity theorem turned out to be true, then by definition there cannot be a solution that contradicts Fermat’s last theorem, which means it should have been proven soon.
Although both theorems were difficult problems in mathematics, considered unsolvable, the work of the two Japanese was the first suggestion of how Fermat's last theorem could be extended and proven for all numbers, not just some. Important to the researchers who chose the research topic was the fact that, unlike Fermat's last theorem, the modularity theorem was a major active area of research for which a proof had been developed, and not just a historical oddity, so the time spent working on it could be justified from a professional point of view. However, the general consensus was that solving the Taniyama-Shimura conjecture was not practical.
Fermat's Last Theorem: Wiles' Proof
After learning that Ribet had proven Frey's theory correct, English mathematician Andrew Wiles, who had been interested in Fermat's last theorem since childhood and had experience working with elliptic curves and related fields, decided to try to prove the Taniyama-Shimura conjecture as a way to prove Fermat's last theorem. In 1993, six years after announcing his goal, while secretly working on the problem of solving the theorem, Wiles managed to prove a related conjecture, which in turn would help him prove Fermat's last theorem. Wiles' document was enormous in size and scope.
The flaw was discovered in one part of his original paper during peer review and required another year of collaboration with Richard Taylor to jointly solve the theorem. As a result, Wiles' final proof of Fermat's Last Theorem was not long in coming. In 1995, it was published on a much smaller scale than Wiles's previous mathematical work, clearly showing that he was not mistaken in his previous conclusions about the possibility of proving the theorem. Wiles' achievement was widely reported in the popular press and popularized in books and television programs. The remaining parts of the Taniyama-Shimura-Weil conjecture, which have now been proven and are known as the modularity theorem, were subsequently proven by other mathematicians who built on Wiles' work between 1996 and 2001. For his achievement, Wiles was honored and received numerous awards, including the 2016 Abel Prize.
Wiles's proof of Fermat's last theorem is a special case of a solution to the modularity theorem for elliptic curves. However, this is the most famous case of such a large-scale mathematical operation. Along with solving Ribet's theorem, the British mathematician also obtained a proof of Fermat's last theorem. Fermat's Last Theorem and the Modularity Theorem were almost universally considered unprovable by modern mathematicians, but Andrew Wiles was able to prove to the entire scientific world that even pundits can be mistaken.
Wiles first announced his discovery on Wednesday 23 June 1993 in a lecture at Cambridge entitled "Modular Forms, Elliptic Curves and Galois Representations". However, in September 1993 it was determined that his calculations contained an error. A year later, on September 19, 1994, in what he would call "the most important moment of his working life," Wiles stumbled upon a revelation that allowed him to correct the solution to the problem to the point where it could satisfy the mathematical community.
Characteristics of work
Andrew Wiles's proof of Fermat's theorem uses many techniques from algebraic geometry and number theory and has many ramifications in these areas of mathematics. He also uses standard constructs of modern algebraic geometry, such as the category of schemes and Iwasawa theory, as well as other 20th-century methods that were not available to Pierre Fermat.
The two articles containing the evidence total 129 pages and were written over seven years. John Coates described this discovery as one of the greatest achievements of number theory, and John Conway called it the main mathematical achievement of the 20th century. Wiles, in order to prove Fermat's last theorem by proving the modularity theorem for the special case of semistable elliptic curves, developed powerful methods for lifting modularity and discovered new approaches to numerous other problems. For solving Fermat's last theorem he was knighted and received other awards. When it was announced that Wiles had won the Abel Prize, the Norwegian Academy of Sciences described his achievement as "a marvelous and elementary proof of Fermat's last theorem."
How it was
One of the people who analyzed Wiles' original manuscript of the theorem's solution was Nick Katz. During his review, he asked the Briton a series of clarifying questions, which forced Wiles to admit that his work clearly contained a gap. There was an error in one critical part of the proof that gave an estimate for the order of a particular group: the Euler system used to extend the Kolyvagin and Flach method was incomplete. The mistake, however, did not render his work useless - each part of Wiles' work was very significant and innovative in itself, as were many of the developments and methods that he created in the course of his work and which affected only one part of the manuscript. However, this original work, published in 1993, did not actually provide a proof of Fermat's Last Theorem.
Wiles spent almost a year trying to rediscover the solution to the theorem, first alone and then in collaboration with his former student Richard Taylor, but all seemed to be in vain. By the end of 1993, rumors had spread that Wiles's proof had failed in testing, but how serious the failure was was not known. Mathematicians began to put pressure on Wiles to reveal the details of his work, whether it was completed or not, so that the wider community of mathematicians could explore and use everything he had achieved. Instead of quickly correcting his mistake, Wiles only discovered additional complexities in the proof of Fermat's last theorem, and finally realized how difficult it was.
Wiles states that on the morning of September 19, 1994, he was on the verge of giving up and giving up, and almost resigned himself to the fact that he had failed. He was willing to publish his unfinished work so that others could build on it and find where he had gone wrong. The English mathematician decided to give himself one last chance and analyzed the theorem one last time to try to understand the main reasons why his approach did not work, when he suddenly realized that the Kolyvagin-Flac approach would not work until he also included proof in the process Iwasawa's theory, making it work.
On October 6, Wiles asked three colleagues (including Faltins) to review his new work, and on October 24, 1994, he submitted two manuscripts, "Modular elliptic curves and Fermat's last theorem" and "Theoretical properties of the ring of some Hecke algebras", the second of which Wiles co-wrote with Taylor and argued that certain conditions necessary to justify the corrected step in the main article were met.
These two papers were reviewed and finally published as a full-text edition in the May 1995 issue of the Annals of Mathematics. Andrew's new calculations were widely analyzed and eventually accepted by the scientific community. These works established the modularity theorem for semistable elliptic curves, the final step towards proving Fermat's Last Theorem, 358 years after it was created.
History of the Great Problem
Solving this theorem has been considered the biggest problem in mathematics for many centuries. In 1816 and again in 1850, the French Academy of Sciences offered a prize for a general proof of Fermat's last theorem. In 1857, the Academy awarded 3,000 francs and a gold medal to Kummer for his research into ideal numbers, although he did not apply for the prize. Another prize was offered to him in 1883 by the Brussels Academy.
Wolfskehl Prize
In 1908, the German industrialist and amateur mathematician Paul Wolfskehl bequeathed 100,000 gold marks (a large sum for that time) to the Göttingen Academy of Sciences as a prize for a complete proof of Fermat's Last Theorem. On June 27, 1908, the Academy published nine awards rules. Among other things, these rules required publication of the evidence in a peer-reviewed journal. The prize was not to be awarded until two years after publication. The competition was due to expire on September 13, 2007 - approximately a century after it began. On June 27, 1997, Wiles received Wolfschel's prize money and then another $50,000. In March 2016, he received €600,000 from the Norwegian government as part of the Abel Prize for his "stunning proof of Fermat's last theorem using the modularity conjecture for semistable elliptic curves, opening a new era in number theory." It was a world triumph for the humble Englishman.
Before Wiles's proof, Fermat's theorem, as mentioned earlier, was considered absolutely unsolvable for centuries. Thousands of incorrect evidence were presented to Wolfskehl's committee at various times, amounting to approximately 10 feet (3 meters) of correspondence. In the first year of the prize's existence alone (1907-1908), 621 applications were submitted claiming to solve the theorem, although by the 1970s this number had decreased to approximately 3-4 applications per month. According to F. Schlichting, Wolfschel's reviewer, most of the evidence was based on rudimentary methods taught in schools and was often presented by "people with a technical background but an unsuccessful career." According to the historian of mathematics Howard Aves, Fermat's last theorem set a kind of record - it is the theorem with the most incorrect proofs.
Fermat laurels went to the Japanese
As mentioned earlier, around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama discovered a possible connection between two apparently completely different branches of mathematics - elliptic curves and modular forms. The resulting modularity theorem (then known as the Taniyama-Shimura conjecture) from their research states that every elliptic curve is modular, meaning that it can be associated with a unique modular form.
The theory was initially dismissed as unlikely or highly speculative, but was taken more seriously when number theorist Andre Weyl found evidence to support the Japanese's findings. As a result, the conjecture was often called the Taniyama-Shimura-Weil conjecture. It became part of the Langlands program, which is a list of important hypotheses that require proof in the future.
Even after serious attention, the conjecture was recognized by modern mathematicians as extremely difficult or perhaps impossible to prove. Now it is this theorem that is waiting for Andrew Wiles, who could surprise the whole world with its solution.
Fermat's theorem: Perelman's proof
Despite the popular myth, the Russian mathematician Grigory Perelman, for all his genius, has nothing to do with Fermat’s theorem. Which, however, does not in any way detract from his numerous services to the scientific community.
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 anyone with a secondary education. It says that the formula a to the power of n + b to the power of n = c to the power of n does not have natural (that is, not fractional) solutions for n > 2. Everything seems simple and clear, but the best mathematicians and ordinary amateurs struggled with searching for a solution for more than three and a half centuries.
Why is she so famous? Now we'll find out...
Are there many proven, unproven and as yet unproven theorems? The point here is that Fermat's Last Theorem represents the greatest contrast between the simplicity of the formulation and the complexity of the proof. Fermat's Last Theorem is an incredibly difficult problem, and yet its formulation can be understood by anyone with the 5th grade of high school, but not even every professional mathematician can understand the proof. Neither in physics, nor in chemistry, nor in biology, nor in mathematics, is there a single problem that could be formulated so simply, but remained unsolved 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 are equal on all sides.” The problem looks so simple because it was based on a mathematical statement that everyone knows - the Pythagorean theorem: in any right triangle, the square built on the hypotenuse is equal to the sum of the squares built on the legs.
In the 5th century BC. Pythagoras founded the Pythagorean brotherhood. The Pythagoreans, among other things, studied integer triplets satisfying the equality x²+y²=z². They proved that there are infinitely many Pythagorean triples and obtained general formulas for finding them. They probably tried to look for C's and higher degrees. Convinced that this did not work, the Pythagoreans abandoned their useless attempts. The members of the brotherhood were more philosophers and aesthetes than mathematicians.
That is, it is easy to select a set of numbers that perfectly satisfy the equality x²+y²=z²
Starting from 3, 4, 5 - indeed, a junior student understands that 9 + 16 = 25.
Or 5, 12, 13: 25 + 144 = 169. Great.
And so on. What if we take a similar equation x³+y³=z³? Maybe there are such numbers too?
And so on (Fig. 1).
So, it turns out that they are NOT. This is where the trick begins. Simplicity is apparent, because it is difficult to prove not the presence of something, but, on the contrary, its absence. When you need to prove that there is a solution, you can and should simply present this solution.
Proving absence is more difficult: 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 solution). And that’s it, the opponent is defeated. How to prove absence?
Say: “I haven’t found such solutions”? Or maybe you weren't looking well? What if they exist, only very large, very large, such that even a super-powerful computer still doesn’t have enough strength? This is what is difficult.
This can be shown visually like this: if you take two squares of suitable sizes and disassemble them into unit squares, then from this bunch of unit squares you get a third square (Fig. 2):
But let’s do the same with the third dimension (Fig. 3) – it doesn’t work. There are not enough cubes, or there are extra ones left:
But the 17th century French mathematician Pierre de Fermat enthusiastically studied the general equation x n +y n =z n . And finally, I concluded: for n>2 there are no integer solutions. Fermat's proof is irretrievably lost. Manuscripts are burning! 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 making mistakes. Even if he did not leave evidence of a statement, it was subsequently confirmed. Moreover, Fermat proved his thesis for n=4. Thus, 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 a proof (in 1770 he proposed a solution for n = 3),
Adrien Legendre and Johann Dirichlet (these scientists jointly found the proof for n = 5 in 1825), Gabriel Lamé (who found the proof for n = 7) and many others. By the mid-80s of the last century, it became clear that the scientific world was on the way to the final solution of Fermat’s Last Theorem, but only in 1993 mathematicians saw and believed that the three-century epic of searching for a proof of Fermat’s last theorem was practically over.
It is easily shown that it is enough to prove Fermat’s theorem only for simple n: 3, 5, 7, 11, 13, 17, ... For composite n, the proof remains valid. But there are infinitely many prime numbers...
In 1825, using the method of Sophie Germain, female mathematicians, Dirichlet and Legendre independently proved the theorem for n=5. In 1839, using the same method, the Frenchman Gabriel Lame showed the truth of the theorem for n=7. Gradually the theorem was proven for almost all n less than one hundred.
Finally, the German mathematician Ernst Kummer, in a brilliant study, showed that the theorem in general cannot be proven using the methods of mathematics of the 19th century. The Prize of the French Academy of Sciences, established in 1847 for the proof of Fermat's theorem, remained unawarded.
In 1907, the wealthy German industrialist Paul Wolfskehl decided to take his own life because of unrequited love. Like a true German, he set the date and time of suicide: exactly at midnight. On the last day he made a will and wrote letters to friends and relatives. Things ended before midnight. It must be said that Paul was interested in mathematics. Having nothing else to do, he went to the library and began to read Kummer’s famous article. Suddenly it seemed to him that Kummer had made a mistake in his reasoning. Wolfskel began to analyze this part of the article with a pencil in his hands. Midnight has passed, morning has come. The gap in the proof has been filled. And the very reason for suicide now looked completely ridiculous. Paul tore up his farewell letters and rewrote his will.
He soon died of natural causes. The heirs were quite surprised: 100,000 marks (more than 1,000,000 current pounds sterling) were transferred to the account of the Royal Scientific Society of Göttingen, which in the same year announced a competition for the Wolfskehl Prize. 100,000 marks were awarded to the person who proved Fermat's theorem. Not a pfennig was awarded for refuting the theorem...
Most professional mathematicians considered the search for a proof of Fermat's Last Theorem a hopeless task and resolutely refused to waste time on such a useless exercise. But the amateurs had a blast. A few weeks after the announcement, an avalanche of “evidence” hit the University of Göttingen. Professor E.M. Landau, whose responsibility was to analyze the evidence sent, distributed cards to his students:
Dear. . . . . . . .
Thank you for sending me the manuscript with the proof of Fermat’s Last Theorem. The first error is on page ... in line... . Because of it, the entire proof loses its validity.
Professor E. M. Landau
In 1963, Paul Cohen, relying 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 undecidable?! But true Great Theorem fanatics were not disappointed at all. The advent of computers suddenly gave mathematicians a new method of proof. After World War II, teams 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 1980s, Samuel Wagstaff raised the limit to 25,000, and in the 1990s, mathematicians declared that Fermat's Last Theorem was true for all values of n up to 4 million. But if you subtract even a trillion trillion from infinity, it will not become smaller. Mathematicians are not convinced by statistics. To prove the Great Theorem meant to prove it for ALL n going to infinity.
In 1954, two young Japanese mathematician friends began researching modular forms. These forms generate series of numbers, each with its own series. By chance, Taniyama compared these series with series generated by elliptic equations. They matched! But modular forms are geometric objects, and elliptic equations are algebraic. No connection has ever been found between such different objects.
However, 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 an entire direction in mathematics, but until the Taniyama-Shimura hypothesis was proven, the entire 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 could not have a counterpart in the modular world. From now on, Fermat's Last Theorem was inextricably linked with the Taniyama–Shimura conjecture. Having proven 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 proven. But for thirty years it was not possible to prove the Taniyama-Shimura hypothesis, and there was less and less hope for success.
In 1963, when he was just ten years old, Andrew Wiles was already fascinated by mathematics. When he learned about the Great Theorem, he realized that he could not give up on it. As a schoolboy, student, and graduate student, he prepared himself for this task.
Having learned about Ken Ribet's findings, Wiles plunged headlong into proving the Taniyama-Shimura conjecture. He decided to work in complete isolation and secrecy. “I realized that everything that had anything to do with Fermat’s Last Theorem arouses too much interest... Too many spectators obviously 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 paper 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. Every piece of evidence must be carefully examined before the evidence can be considered rigorous and accurate. Wiles spent a restless summer waiting for feedback from reviewers, hoping that he would be able to win their approval. At the end of August, experts found the judgment to be insufficiently substantiated.
It turned out that this decision contains a gross error, although in general it is correct. Wiles did not give up, called on the help of the famous specialist in number theory Richard Taylor, and already in 1994 they published a corrected and expanded proof of the theorem. The most amazing thing is that this work took up as many as 130 (!) pages in the mathematical journal “Annals of Mathematics”. But the story did not end there either - the final point was reached only in the next 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 presented Nadya with the manuscript of the complete proof” (Andrew Wales). Have I not yet said that mathematicians are strange people?
This time there was no doubt about the evidence. Two articles were subjected to the most careful analysis and were published in May 1995 in the Annals of Mathematics.
A lot of time has passed since that moment, but there is still an opinion in society that Fermat’s Last Theorem is unsolvable. But even those who know about the proof found continue to work in this direction - few are satisfied that the Great Theorem requires a solution of 130 pages!
Therefore, now the efforts of many mathematicians (mostly amateurs, not professional scientists) are thrown into the search for a simple and concise proof, but this path, most likely, will not lead anywhere...
For integers n greater than 2, the equation x n + y n = z n has no nonzero solutions in natural numbers.
You probably remember from your school days Pythagorean theorem: The square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs. You may also remember the classic right triangle with sides whose lengths are in the ratio 3: 4: 5. For it, the Pythagorean theorem looks like this:
This is an example of solving the generalized Pythagorean equation in non-zero integers for n= 2. Fermat's Last Theorem (also called "Fermat's Last Theorem" and "Fermat's Last Theorem") is the statement that for the values n> 2 equations of the form x n + y n = z n have no non-zero solutions in natural numbers.
The history of Fermat's Last Theorem is very interesting and instructive, and not only for mathematicians. Pierre de Fermat contributed to the development of various fields of mathematics, but the main part of his scientific legacy was published only posthumously. The fact is that mathematics for Fermat was something of a hobby, and not a professional occupation. He corresponded with the leading mathematicians of his time, but did not strive to publish his work. Fermat's scientific writings are mainly found in the form of private correspondence and fragmentary notes, often written in the margins of various books. It is in the margins (of the second volume of the ancient Greek “Arithmetic” of Diophantus. - Note translator) soon after the death of the mathematician, the descendants discovered the formulation of the famous theorem and the postscript:
« I found a truly wonderful proof of this, but these fields are too narrow for it».
Alas, apparently, Fermat never bothered to write down the “miraculous proof” he found, and descendants unsuccessfully searched for it for more than three centuries. Of all Fermat's scattered scientific heritage, which contains many surprising statements, it was the Great Theorem that stubbornly refused to be solved.
Whoever has tried to prove Fermat's Last Theorem is in vain! Another great French mathematician, René Descartes (1596–1650), called Fermat a “braggart,” and the English mathematician John Wallis (1616–1703) called him a “damn Frenchman.” Fermat himself, however, still left behind a proof of his theorem for the case n= 4. With proof for n= 3 was solved by the great Swiss-Russian mathematician of the 18th century Leonhard Euler (1707–83), after which, unable to find evidence for n> 4, jokingly suggested that Fermat's house be searched to find the key to the lost evidence. In the 19th century, new methods in number theory made it possible to prove the statement for many integers within 200, but again, not for all.
In 1908, a prize of 100,000 German marks was established for solving this problem. The prize fund was bequeathed by the German industrialist Paul Wolfskehl, who, according to legend, was going to commit suicide, but was so carried away by Fermat's Last Theorem that he changed his mind about dying. With the advent of adding machines and then computers, the value bar n began to rise higher and higher - to 617 by the beginning of World War II, to 4001 in 1954, to 125,000 in 1976. At the end of the 20th century, the most powerful computers at military laboratories in Los Alamos (New Mexico, USA) were programmed to solve Fermat's problem in the background (similar to the screen saver mode of a personal computer). Thus, it was possible to show that the theorem is true for incredibly large values x, y, z And n, but this could not serve as a strict proof, since any of the following values n or triplets of natural numbers could disprove the theorem as a whole.
Finally, in 1994, the English mathematician Andrew John Wiles (b. 1953), working at Princeton, published a proof of Fermat's Last Theorem, which, after some modifications, was considered comprehensive. The proof took more than a hundred journal pages and was based on the use of modern apparatus of higher mathematics, which was not developed in Fermat’s era. So what then did Fermat mean by leaving a message in the margins of the book that he had found the proof? Most of the mathematicians with whom I spoke on this topic pointed out that over the centuries there had been more than enough incorrect proofs of Fermat's Last Theorem, and that, most likely, Fermat himself had found a similar proof, but failed to recognize the error in it. However, it is possible that there is still some short and elegant proof of Fermat’s Last Theorem that no one has yet found. Only one thing can be said with certainty: today we know for sure that the theorem is true. Most mathematicians, I think, would agree unreservedly with Andrew Wiles, who remarked of his proof: “Now at last my mind is at peace.”
