The Great Farm Theorem. I want to study - unsolved problems Theories that have not been proven
Lev Valentinovich Rudi, the author of the article "Pierre Fermat and his "unprovable" theorem", having read a publication about one of the 100 geniuses of modern mathematics, who was called a genius due to his solution of Fermat's theorem, offered to publish his alternative opinion on this topic. To which we readily responded and publish his article without abbreviations.
Pierre de Fermat and his "unprovable" theorem
This year marks the 410th anniversary of the birth of the great French mathematician Pierre de Fermat. Academician V.M. Tikhomirov writes about P. Fermat: “Only one mathematician has been honored with the fact that his name has become a household name. If they say "fermatist", then we are talking about a person obsessed to the point of insanity by some unrealizable idea. But this word cannot be attributed to Pierre Fermat (1601-1665), one of the brightest minds in France, himself.
P. Fermat is a man of amazing destiny: one of the greatest mathematicians in the world, he was not a "professional" mathematician. Fermat was a lawyer by profession. He received an excellent education and was an outstanding connoisseur of art and literature. All his life he worked in the civil service, for the last 17 years he was an adviser to the parliament in Toulouse. A disinterested and sublime love attracted him to mathematics, and it was this science that gave him everything that love can give a person: intoxication with beauty, pleasure and happiness.
In papers and correspondence, Fermat formulated many beautiful statements, about which he wrote that he had their proof. And gradually there were fewer and fewer such unproven statements and, finally, only one remained - his mysterious Great Theorem!
However, for those interested in mathematics, Fermat's name speaks volumes regardless of his Grand Theorem. He was one of the most insightful minds of his time, he is considered the founder of number theory, he made a huge contribution to the development of analytic geometry, mathematical analysis. We are grateful to Fermat for opening for us a world full of beauty and mystery” (nature.web.ru:8001›db/msg.html…).
Strange, however, "gratitude"!? The mathematical world and enlightened humanity ignored Fermat's 410th anniversary. Everything was, as always, quiet, peaceful, everyday ... There was no fanfare, laudatory speeches, toasts. Of all the mathematicians in the world, only Fermat was “honored” with such a high honor that when the word “fermatist” is used, everyone understands that we are talking about a half-wit who is “madly obsessed with an unrealizable idea” to find the lost proof of Fermat's theorem!
In his remark on the margin of Diophantus's book, Fermas wrote: "I have found a truly amazing proof of my assertion, but the margins of the book are too narrow to accommodate it." So it was "the moment of weakness of the mathematical genius of the 17th century." This dumbass did not understand that he was “mistaken”, but, most likely, he simply “lied”, “cunning”.
If Fermat claimed, then he had proof!? The level of knowledge was no higher than that of a modern tenth grader, but if some engineer tries to find this proof, then he is ridiculed, declared insane. And it is a completely different matter if an American 10-year-old boy E. Wiles "accepts as an initial hypothesis that Fermat could not know much more mathematics than he does" and begins to "prove" this "unprovable theorem." Of course, only a “genius” is capable of such a thing.
By chance, I came across a site (works.tarefer.ru›50/100086/index.html), where a student of the Chita State Technical University Kushenko V.V. writes about Fermat: “... The small town of Beaumont and all its five thousand inhabitants are unable to realize that the great Fermat was born here, the last mathematician-alchemist who solved the idle problems of the coming centuries, the quietest judicial hook, the crafty sphinx who tortured humanity with its riddles , a cautious and virtuous bureaucrat, a swindler, an intriguer, a homebody, an envious person, a brilliant compiler, one of the four titans of mathematics ... Farm almost never left Toulouse, where he settled after marrying Louise de Long, the daughter of an adviser to parliament. Thanks to his father-in-law, he rose to the rank of adviser and acquired the coveted prefix "de". The son of the third estate, the practical offspring of wealthy leather workers, stuffed with Latin and Franciscan piety, he did not set himself grandiose tasks in real life ...
In his turbulent age, he lived thoroughly and quietly. He did not write philosophical treatises, like Descartes, was not the confidant of the French kings, like Viet, did not fight, did not travel, did not create mathematical circles, did not have students and was not published during his lifetime ... Having found no conscious claims to a place in history, The farm dies on January 12, 1665."
I was shocked, shocked... And who was the first "mathematician-alchemist"!? What are these “idle tasks of the coming centuries”!? “A bureaucrat, a swindler, an intriguer, a homebody, an envious person” ... Why do these green youths and youths have so much disdain, contempt, cynicism for a person who lived 400 years before them!? What blasphemy, blatant injustice!? But, not the youngsters themselves came up with all this!? They were thought up by mathematicians, "kings of sciences", that same "humanity", which Fermat's "cunning sphinx" "tortured with his riddles".
However, Fermat cannot bear any responsibility for the fact that arrogant, but mediocre descendants for more than three hundred years knocked their horns on his school theorem. Humiliating, spitting on Fermat, mathematicians are trying to save their honor of uniform!? But there has been no “honor” for a long time, not even a “uniform”!? Fermat's children's problem has become the greatest shame of the "selected, valiant" army of mathematicians of the world!?
The “kings of sciences” were disgraced by the fact that seven generations of mathematical “luminaries” could not prove the school theorem, which was proved by both P. Fermat and the Arab mathematician al-Khujandi 700 years before Fermat!? They were also disgraced by the fact that, instead of admitting their mistakes, they denounced P. Fermat as a deceiver and began to inflate the myth about the “unprovability” of his theorem!? Mathematicians have also disgraced themselves by the fact that for a whole century they have been frenziedly persecuting amateur mathematicians, "beating their smaller brothers on the head." This persecution became the most shameful act of mathematicians in the entire history of scientific thought after the drowning of Hippasus by Pythagoras! They were also disgraced by the fact that, under the guise of a "proof" of Fermat's theorem, they slipped enlightened humanity the dubious "creation" of E. Wiles, which even the brightest luminaries of mathematics "do not understand"!?
The 410th anniversary of the birth of P. Fermat is undoubtedly a strong enough argument for mathematicians to finally come to their senses and stop casting a shadow on the wattle fence and restore the good, honest name of the great mathematician. P. Fermat “did not find any conscious claims to a place in history,” but this wayward and capricious Lady herself entered it in her annals in her arms, but she spat out many zealous and zealous “applicants” like chewed gum. And nothing can be done about it, just one of his many beautiful theorems forever entered the name of P. Fermat in history.
But this unique creation of Fermat has been driven underground for a whole century, outlawed, and has become the most contemptible and hated task in the entire history of mathematics. But the time has come for this "ugly duckling" of mathematics to turn into a beautiful swan! Fermat's amazing riddle has earned its right to take its rightful place in the treasury of mathematical knowledge, and in every school of the world, next to its sister, the Pythagorean theorem.
Such a unique, elegant problem simply cannot but have beautiful, elegant solutions. If the Pythagorean theorem has 400 proofs, then let Fermat's theorem have only 4 simple proofs at first. They are, gradually there will be more of them!? I believe that the 410th anniversary of P. Fermat is the most suitable occasion or occasion for professional mathematicians to come to their senses and finally stop this senseless, absurd, troublesome and absolutely useless "blockade" of amateurs!?
For integers n greater than 2, the equation x n + y n = z n has no non-zero solutions in natural numbers.
You probably remember from your school days the 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 related as 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 values n> 2 equations of the form x n + y n = z n do not have nonzero solutions in natural numbers.
The history of Fermat's Last Theorem is very entertaining and instructive, and not only for mathematicians. Pierre de Fermat contributed to the development of various areas of mathematics, but the main part of his scientific heritage was published only posthumously. The fact is that mathematics for Fermat was something like a hobby, not a professional occupation. He corresponded with the leading mathematicians of his time, but did not seek to publish his work. Fermat's scientific writings are mostly found in the form of private correspondence and fragmentary notes, often made in the margins of various books. It is on the margins (of the second volume of the ancient Greek Arithmetic by Diophantus. - Note. translator) shortly 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 margins are too narrow for him.».
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 disparate scientific heritage, containing many surprising statements, it was the Great Theorem that stubbornly resisted solution.
Whoever did not take up the proof of Fermat's Last Theorem - all in vain! Another great French mathematician, René Descartes (René Descartes, 1596-1650), called Fermat a "braggart", and the English mathematician John Wallis (John Wallis, 1616-1703) called him a "damn Frenchman". Fermat himself, however, nevertheless 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 Leonard Euler (1707–83), after which, having failed to find proofs for n> 4, jokingly offered to search Fermat's house to find the key to the lost evidence. In the 19th century, new methods of number theory made it possible to prove the statement for many integers within 200, but, again, not for all.
In 1908 a prize of DM 100,000 was established for this task. The prize fund was bequeathed to the German industrialist Paul Wolfskehl, who, according to legend, was about 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 bar of values n began to rise higher and higher - up to 617 by the beginning of World War II, up to 4001 in 1954, up to 125,000 in 1976. At the end of the 20th century, the most powerful computers of military laboratories in Los Alamos (New Mexico, USA) were programmed to solve the Fermat 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 rigorous proof, since any of the following values n or triples of natural numbers could disprove the theorem as a whole.
Finally, in 1994, the English mathematician Andrew John Wiles (Andrew John Wiles, b. 1953), while working at Princeton, published a proof of Fermat's Last Theorem, which, after some modifications, was considered exhaustive. The proof took more than a hundred magazine pages and was based on the use of the modern apparatus of higher mathematics, which had not been 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 proof? Most of the mathematicians I have spoken to on this subject have pointed out that over the centuries there have been more than enough incorrect proofs of Fermat's Last Theorem, and that it is likely that Fermat himself found a similar proof but failed to see the error in it. However, it is possible that there is still some short and elegant proof of Fermat's Last Theorem, which 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 unreservedly agree with Andrew Wiles, who remarked about his proof, "Now at last my mind is at peace."
There are not so many people in the world who have never heard of Fermat's Last Theorem - perhaps this is the only mathematical problem that has received such wide popularity and has become 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 simple 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 5 grades of secondary school, but the proof is far from even every 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 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 triples satisfying the equation 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 triples and higher 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 studied the 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-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 did mathematicians see and believe 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 there are infinitely many prime numbers...
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 of 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 ...
a source
1 Murad :
We considered the equality Zn = Xn + Yn to be the Diophantus equation or Fermat's Great Theorem, and this is the solution of the equation (Zn- Xn) Xn = (Zn - Yn) Yn. Then Zn =-(Xn + Yn) is a solution to the equation (Zn + Xn) Xn = (Zn + Yn) Yn. These equations and solutions are related to the properties of integers and operations on them. So we don't know the properties of integers?! With such limited knowledge, we will not reveal the truth.
Consider the solutions Zn = +(Xn + Yn) and Zn =-(Xn + Yn) when n = 1. Integers + Z are formed using 10 digits: 0, 1, 2, 3, 4, 5, 6, 7 , 8, 9. They are divisible by 2 integers +X - even, last right digits: 0, 2, 4, 6, 8 and +Y - odd, last right digits: 1, 3, 5, 7, 9, t .e. + X = + Y. The number of Y = 5 - odd and X = 5 - even numbers is: Z = 10. Satisfies the equation: (Z - X) X = (Z - Y) Y, and the solution + Z = + X + Y= +(X + Y).
Integers -Z consist of the union of -X for even and -Y for odd, and satisfies the equation:
(Z + X) X = (Z + Y) Y, and the solution -Z = - X - Y = - (X + Y).
If Z/X = Y or Z / Y = X, then Z = XY; Z / -X = -Y or Z / -Y = -X, then Z = (-X)(-Y). Division is checked by multiplication.
Single-digit positive and negative numbers consist of 5 odd and 5 odd numbers.
Consider the case n = 2. Then Z2 = X2 + Y2 is a solution to the equation (Z2 – X2) X2 = (Z2 – Y2) Y2 and Z2 = -(X2 + Y2) is a solution to the equation (Z2 + X2) X2 = (Z2 + Y2) Y2. We considered Z2 = X2 + Y2 to be the Pythagorean theorem, and then the solution Z2 = -(X2 + Y2) is the same theorem. We know that the diagonal of a square divides it into 2 parts, where the diagonal is the hypotenuse. Then the equalities are valid: Z2 = X2 + Y2, and Z2 = -(X2 + Y2) where X and Y are legs. And more solutions R2 = X2 + Y2 and R2 =- (X2 + Y2) are circles, centers are the origin of the square coordinate system and with radius R. They can be written as (5n)2 = (3n)2 + (4n)2 , where n are positive and negative integers, and are 3 consecutive numbers. Also solutions are 2-bit XY numbers that starts at 00 and ends at 99 and is 102 = 10x10 and count 1 century = 100 years.
Consider solutions when n = 3. Then Z3 = X3 + Y3 are solutions of the equation (Z3 – X3) X3 = (Z3 – Y3) Y3.
3-bit numbers XYZ starts at 000 and ends at 999 and is 103 = 10x10x10 = 1000 years = 10 centuries
From 1000 cubes of the same size and color, you can make a rubik of about 10. Consider a rubik of the order +103=+1000 - red and -103=-1000 - blue. They consist of 103 = 1000 cubes. If we decompose and put the cubes in one row or on top of each other, without gaps, we get a horizontal or vertical segment of length 2000. Rubik is a large cube, covered with small cubes, starting from the size 1butto = 10st.-21, and you cannot add to it or subtract one cube.
- (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9+10); + (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9+10);
- (12 + 22 + 32 + 42 + 52 + 62 + 72 + 82 + 92+102); + (12 + 22 + 32 + 42 + 52 + 62 + 72 + 82 + 92+102);
- (13 + 23 + 33 + 43 + 53 + 63 + 73 + 83 + 93+103); + (13 + 23 + 33 + 43 + 53 + 63 + 73 + 83 + 93+103).
Each integer is 1. Add 1(ones) 9 + 9 =18, 10 + 9 =19, 10 +10 =20, 11 +10 =21, and the products:
111111111 x 111111111 = 12345678987654321; 1111111111 x 111111111 = 123456789987654321.
0111111111x1111111110= 0123456789876543210; 01111111111x1111111110= 01234567899876543210.
These operations can be performed on 20-bit calculators.
It is known that +(n3 - n) is always divisible by +6, and - (n3 - n) is divisible by -6. We know that n3 - n = (n-1)n(n+1). This is 3 consecutive numbers (n-1)n(n+1), where n is even, then divisible by 2, (n-1) and (n+1) odd, divisible by 3. Then (n-1) n(n+1) is always divisible by 6. If n=0, then (n-1)n(n+1)=(-1)0(+1), n=20, then(n-1)n (n+1)=(19)(20)(21).
We know that 19 x 19 = 361. This means that one square is surrounded by 360 squares, and then one cube is surrounded by 360 cubes. The equality is fulfilled: 6 n - 1 + 6n. If n=60, then 360 - 1 + 360, and n=61, then 366 - 1 + 366.
The following generalizations follow from the above statements:
n5 - 4n = (n2-4) n (n2+4); n7 - 9n = (n3-9) n (n3+9); n9 –16 n= (n4-16) n (n4+16);
0… (n-9) (n-8) (n-7) (n-6) (n-5) (n-4) (n-3) (n-2) (n-1)n(n +1) (n+2) (n+3) (n+4) (n+5) (n+6) (n+7) (n+8) (n+9)…2n
(n+1) x (n+1) = 0123… (n-3) (n-2) (n-1) n (n+1) n (n-1) (n-2) (n-3 )…3210
n! = 0123… (n-3) (n-2) (n-1) n; n! = n (n-1) (n-2) (n-3)…3210; (n+1)! =n! (n+1).
0 +1 +2+3+…+ (n-3) + (n-2) + (n-1) +n=n (n+1)/2; n + (n-1) + (n-2) + (n-3) +…+3+2+1+0=n (n+1)/2;
n (n+1)/2 + (n+1) + n (n+1)/2 = n (n+1) + (n+1) = (n+1) (n+1) = (n +1)2.
If 0123… (n-3) (n-2) (n-1) n (n+1) n (n-1) (n-2) (n-3)…3210 x 11=
= 013… (2n-5) (2n-3) (2n-1) (2n+1) (2n+1) (2n-1) (2n-3) (2n-5)…310.
Any integer n is a power of 10, has: – n and +n, +1/ n and -1/ n, odd and even:
- (n + n +…+ n) = -n2; – (n x n x…x n) = -nn; – (1/n + 1/n +…+ 1/n) = – 1; – (1/n x 1/n x…x1/n) = -n-n;
+ (n + n +…+ n) =+n2; + (n x n x…x n) = + nn; + (1/n +…+1/n) = + 1; + (1/n x 1/n x…x1/n) = + n-n.
It is clear that if any integer is added to itself, then it will increase by 2 times, and the product will be a square: X = a, Y = a, X + Y = a + a = 2a; XY = a x a = a2. This was considered Vieta's theorem - a mistake!
If we add and subtract the number b to the given number, then the sum does not change, but the product changes, for example:
X \u003d a + b, Y \u003d a - b, X + Y \u003d a + b + a - b \u003d 2a; XY \u003d (a + b) x (a -b) \u003d a2-b2.
X = a +√b, Y = a -√b, X+Y = a +√b + a – √b = 2a; XY \u003d (a + √b) x (a - √b) \u003d a2- b.
X = a + bi, Y = a - bi, X + Y = a + bi + a - bi = 2a; XY \u003d (a + bi) x (a -bi) \u003d a2 + b2.
X = a + √b i, Y = a - √bi, X+Y = a + √bi+ a - √bi =2a, XY = (a -√bi) x (a -√bi) = a2+b.
If we put integer numbers instead of letters a and b, then we get paradoxes, absurdities, and mistrust of mathematics.
Often, when talking with high school students about research work in mathematics, I hear the following: "What new things can be discovered in mathematics?" But really: maybe all the great discoveries have been made, and the theorems have been proven?
On August 8, 1900, at the International Congress of Mathematicians in Paris, mathematician David Hilbert outlined a list of problems that he believed were to be solved in the twentieth century. There were 23 items on the list. Twenty-one of them have been resolved so far. The last solved problem on Gilbert's list was Fermat's famous theorem, which scientists couldn't solve for 358 years. In 1994, the Briton Andrew Wiles proposed his solution. It turned out to be true.Following the example of Gilbert at the end of the last century, many mathematicians tried to formulate similar strategic tasks for the 21st century. One such list was made famous by Boston billionaire Landon T. Clay. In 1998, at his expense, the Clay Mathematics Institute was founded in Cambridge (Massachusetts, USA) and prizes were established for solving a number of important problems in modern mathematics. On May 24, 2000, the institute's experts chose seven problems - according to the number of millions of dollars allocated for prizes. The list is called the Millennium Prize Problems:
1. Cook's problem (formulated in 1971)
Let's say that you, being in a large company, want to make sure that your friend is also there. If you are told that he is sitting in the corner, then a fraction of a second will be enough to, with a glance, make sure that the information is true. In the absence of this information, you will be forced to go around the entire room, looking at the guests. This suggests that solving a problem often takes more time than checking the correctness of the solution.
Stephen Cook formulated the problem: can checking the correctness of a solution to a problem be longer than getting the solution itself, regardless of the verification algorithm. This problem is also one of the unsolved problems in the field of logic and computer science. Its solution could revolutionize the fundamentals of cryptography used in the transmission and storage of data.
2. The Riemann Hypothesis (formulated in 1859)
Some integers cannot be expressed as the product of two smaller integers, such as 2, 3, 5, 7, and so on. Such numbers are called prime numbers and play an important role in pure mathematics and its applications. The distribution of prime numbers among the series of all natural numbers does not follow any regularity. However, the German mathematician Riemann made an assumption regarding the properties of a sequence of prime numbers. If the Riemann Hypothesis is proven, it will revolutionize our knowledge of encryption and lead to unprecedented breakthroughs in Internet security.
3. Birch and Swinnerton-Dyer hypothesis (formulated in 1960)
Associated with the description of the set of solutions of some algebraic equations in several variables with integer coefficients. An example of such an equation is the expression x2 + y2 = z2. Euclid gave a complete description of the solutions to this equation, but for more complex equations, finding solutions becomes extremely difficult.
4. Hodge hypothesis (formulated in 1941)
In the 20th century, mathematicians discovered a powerful method for studying the shape of complex objects. The main idea is to use simple "bricks" instead of the object itself, which are glued together and form its likeness. The Hodge hypothesis is connected with some assumptions about the properties of such "bricks" and objects.
5. The Navier - Stokes equations (formulated in 1822)
If you sail in a boat on the lake, then waves will arise, and if you fly in an airplane, turbulent currents will arise in the air. It is assumed that these and other phenomena are described by equations known as the Navier-Stokes equations. The solutions of these equations are unknown, and it is not even known how to solve them. It is necessary to show that the solution exists and is a sufficiently smooth function. The solution of this problem will make it possible to significantly change the methods of carrying out hydro- and aerodynamic calculations.
6. Poincare problem (formulated in 1904)
If you stretch a rubber band over an apple, then you can slowly move the tape without leaving the surface, compress it to a point. On the other hand, if the same rubber band is properly stretched around the donut, there is no way to compress the band to a point without tearing the band or breaking the donut. The surface of an apple is said to be simply connected, but the surface of a donut is not. It turned out to be so difficult to prove that only the sphere is simply connected that mathematicians are still looking for the correct answer.
7. Yang-Mills equations (formulated in 1954)
The equations of quantum physics describe the world of elementary particles. Physicists Yang and Mills, having discovered the connection between geometry and elementary particle physics, wrote their own equations. Thus, they found a way to unify the theories of electromagnetic, weak and strong interactions. From the Yang-Mills equations, the existence of particles followed, which were actually observed in laboratories all over the world, therefore the Yang-Mills theory is accepted by most physicists, despite the fact that within this theory it is still not possible to predict the masses of elementary particles.
I think that this material published on the blog is interesting not only for students, but also for schoolchildren who are seriously involved in mathematics. There is something to think about when choosing topics and areas of research.
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