I want to study - unsolved problems. Math I like Unsolved math problems three circles
Often, 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 Full description solutions of 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. The Yang-Mills equations implied the existence of particles that were indeed observed in laboratories all over the world, so the Yang-Mills theory is accepted by most physicists, despite the fact that this theory still fails 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. Fermat's interest in mathematics appeared somehow unexpectedly and at a fairly mature age. In 1629, a Latin translation of Pappus's work, containing a brief summary of Apollonius' results on the properties of conic sections, fell into his hands. Fermat, a polyglot, an expert in law and ancient philology, suddenly sets out to completely restore the course of reasoning of the famous scientist. With the same success, a modern lawyer can try to independently reproduce all the proofs from a monograph from problems, say, of algebraic topology. However, the unthinkable enterprise is crowned with success. Moreover, delving into the geometric constructions of the ancients, he makes an amazing discovery: in order to find the maxima and minima of the areas of figures, ingenious drawings are not needed. It is always possible to compose and solve some simple algebraic equation, the roots of which determine the extremum. He came up with an algorithm that would become the basis of differential calculus.
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 many 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 a true center French science. Subsequently, the Mersenne circle by decree 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 - “The method of finding maxima and minima” and “Answers to B. Cavalieri's questions”. 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 “New Solid Geometry”. 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 of 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 the end of the 18th century, Joseph Louis Comte de Lagrange spoke out even more clearly: “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 any special reservations that in my proposal it is required to find integers, since in the case of fractional numbers the most insignificant arithmeticist 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 of letter designations, 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 key concepts 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” word by 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 thing: during this period, his scientific interests changed dramatically. 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 V. 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.
Lev Valentinovich Rudi, the author of the article “Pierre Fermat and his “unprovable” theorem”, after reading 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 for public 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 to 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!?
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.
(Socrates, ancient Greek philosopher)
NOBODY is given to own the universal mind and know EVERYTHING. Nevertheless, most scientists, and even those who simply love to think and explore, always have a desire to learn more, to solve mysteries. But are there still unsolved topics in humanity? After all, it seems that everything is already clear and you just need to apply the knowledge gained over the centuries?
Do NOT despair! There are still unsolved problems from the field of mathematics, logic, which in 2000 the experts of the Clay Mathematical Institute in Cambridge (Massachusetts, USA) combined into a list of the so-called 7 mysteries of the Millennium (Millennium Prize Problems). These problems concern scientists all over the planet. From then to this day, anyone can claim to have found a solution to one of the problems, prove a hypothesis, and receive an award from Boston billionaire Landon Clay (after whom the institute is named). He has already allocated $7 million for this purpose. By the way, Today, one of the problems has already been solved.
So, are you ready to learn about math riddles?
Navier-Stokes equations (formulated in 1822)
Field: hydroaerodynamicsThe equations for turbulent, air, and fluid flows are known as the Navier-Stokes equations. If, for example, you float on a lake on something, then waves will inevitably arise around you. This also applies to airspace: when flying in an airplane, turbulent flows will also form in the air.
These equations just produce description of the processes of motion of a viscous fluid and are the core problem of all hydrodynamics. For some particular cases, solutions have already been found in which parts of the equations are discarded as not affecting the final result, but in general terms, solutions to these equations have not been found.
It is necessary to find a solution to the equations and identify smooth functions.
Riemann hypothesis (formulated in 1859)
Field: number theoryIt is known that the distribution of prime numbers (which are divisible only by themselves and by one: 2,3,5,7,11…) among all natural numbers does not follow any regularity.
The German mathematician Riemann thought about this problem, who made his assumption, theoretically concerning the properties of the existing sequence of prime numbers. The so-called paired prime numbers have long been known - twin prime numbers, the difference between which is 2, for example, 11 and 13, 29 and 31, 59 and 61. Sometimes they form whole clusters, for example, 101, 103, 107, 109 and 113 .
If such accumulations are found and a certain algorithm is derived, this will lead to a revolutionary change in our knowledge in the field of encryption and to an unprecedented breakthrough in the field of Internet security.
Poincare problem (formulated in 1904. Solved in 2002.)
Field: topology or geometry of multidimensional spacesThe essence of the problem lies in the topology and lies in the fact that if you stretch a rubber band, for example, on an apple (sphere), then it will be theoretically possible to compress it to a point, slowly moving the tape without taking it off the surface. However, if the same tape is pulled around a donut (torus), then it is not possible to compress the tape without breaking the tape or breaking the donut itself. Those. the entire surface of a sphere is simply connected, while that of a torus is not. The task was to prove that only the sphere is simply connected.
Representative of the Leningrad Geometric School Grigory Yakovlevich Perelman is the recipient of the Clay Institute of Mathematics Millennium Prize (2010) for solving the Poincaré problem. He refused the famous Fildes Prize.
Hodge hypothesis (formulated in 1941)
Field: algebraic geometryIn reality, there are many simple and much more complex geometric objects. The more complex the object, the more difficult it is to study it. Now scientists have invented and are using with might and main an approach based on the use of parts of one whole ("bricks") to study this object, as an example - a constructor. Knowing the properties of the "bricks", it becomes possible to approach the properties of the object itself. The Hodge hypothesis in this case is connected with some properties of both "bricks" and objects.
This is a very serious problem in algebraic geometry: to find exact ways and methods to analyze complex objects with the help of simple "bricks".
Yang-Mills equations (formulated in 1954)
Field: geometry and quantum physicsPhysicists Yang and Mills describe the world of elementary particles. They, having discovered the connection between geometry and elementary particle physics, wrote their own equations in the field of quantum physics. Thereby a way was found to unify the theories of electromagnetic, weak and strong interactions.
At the level of microparticles, an “unpleasant” effect arises: if several fields act on a particle at once, their combined effect can no longer be decomposed into the action of each of them individually. This is due to the fact that in this theory, not only particles of matter are attracted to each other, but also themselves lines of force fields.
Although the Yang-Mills equations are accepted by all physicists of the world, the theory concerning the prediction of the mass of elementary particles has not been experimentally proven.
Birch and Swinnerton-Dyer hypothesis (formulated in 1960)
Field: algebra and number theoryHypothesis related to the equations of elliptic curves and the set of their rational solutions. In the proof of Fermat's theorem, elliptic curves took one of the important places. And in cryptography, they form a whole section of the name itself, and some Russian digital signature standards are based on them.
The problem is that you need to describe ALL solutions in integers x, y, z of algebraic equations, that is, equations in several variables with integer coefficients.
Cook's problem (formulated in 1971)
Field: mathematical logic and cyberneticsIt is also called "Equality of classes P and NP", and it is one of the most important problems in the theory of algorithms, logic and computer science.
Can the process of checking the correctness of the solution of a problem last longer than the time spent on the solution of this problem itself(regardless of the verification algorithm)?
The solution of the same problem, sometimes, takes a different amount of time, if you change the conditions and algorithms. For example: in a large company you are looking for a friend. If you know that he is sitting in a corner or at a table, then it will take you a split second to see him. But if you do not know exactly where the object is, then spend more time looking for it, bypassing all the guests.
The main question is: can all or not all problems that can be easily and quickly checked be also easily and quickly solved?
Mathematics, as it may seem to many, is not so far from reality. It is the mechanism by which our world and many phenomena can be described. Math is everywhere. And V.O. was right. Klyuchevsky, who said: "It's not the flowers' fault that the blind can't see them".
