Let the first number (I) be s. In order for the square of it *when adding the second number to give a square, the second number must be 2s + 1, since in this case the requirement of the problem is fulfilled: the square of the first number. folded with the second, gives

s2 + 2s + 1, that is, the full square of (s + 1)2.

The square of the second number added to the first must also give a square, that is, the number (2s + I) 2 + s, equal to

4s2 + 5s + 1 == t2

Let us assume that t = 2s -- 2; then t 2 \u003d 4s 2 - 8s + 4. This expression should equal 4s 2 + 5s + 1. So, it should be:

4s 2 -- 8s + 4 == 4s 2 + 5s + l whence s=

So, the numbers satisfy the problem:

Examination;

Why Diophantus makes the assumption that t==2s--2, he does not explain. In all his problems (there are 189 of them in the six books that have come down to us), he makes this or that assumption, without giving any justification.

There are 189 problems in Arithmetic, each with one or more solutions. The tasks are set in a general form, then the specific values ​​of the quantities included in it are taken and solutions are given.

The objectives of Book I are for the most part definite. It also contains those that are solved using systems of two equations with two unknowns, equivalent to a quadratic equation. For its solvability, Diophantus puts forward the condition that the discriminant be a perfect square. So, problem 30 - to find two numbers such that their difference and product are given numbers - is reduced to the system

x - y \u003d a, x \u003d b.

Diophantus puts forward a “formation condition”: it is required that the quadruple product of numbers, added to the square of their difference, be a square, i.e.

4b + a 2 = with 2 .

In book II, problems are solved related to indefinite equations and systems of such equations with 2, 3, 4, 5, 6 unknowns of degree not higher than the second.

Diophantus uses various techniques. Let it be necessary to solve an indefinite equation of the second degree with two unknowns f 2 (x, y)==0. If he has a rational solution (x 0 ,y 0 ), then Diophantus introduces a substitution

wherein k rationally. After that, the main equation is converted into a quadratic one with respect to t, whose free term f 2 (x 0 , y 0 ) = 0. From the equation it turns out t 1 == 0 (Diophantus discards this value), t 2 is a rational number. Then substitution gives rational X and y.

In the case when the problem was reduced to the equation

at 2 = ax 2 + bx + With, obviously rational solution

x 0 = Oh, y 0 =±C. The Diophantus substitution looks like this:

y=kt ± c

Diophantus used another method in solving the problems of book II when they led to the equation at 2 == = a 2 x 2 + bx + With. He made a substitution

then X and at expressed rationally through the parameter k:

Diophantus, in essence, applied the theorem, which consists in; that if an indefinite equation has at least one rational solution, then there will be an infinite number of such solutions, and the values X and at can be represented as rational functions of some parameter"

In book II there are problems solved with the help of the "double inequality", i.e., the system

cx + d == v 2 .

Diophantus considers the case a= c, but subsequently writes that the method can also be applied when a : c = t 2 , When a\u003d\u003d c, Diophantus, by subtracting one equality from another, gets and 2 --and 2 = b -- d. Then the difference b -- d multiplied b -- d = n1 and equates and + v = I, and -- v = n, after which he finds

and \u003d (I + n) / 2, v \u003d (I - n) / 2, x - (l 2 + n 2) / 4a - (b + d) / 2a.

If the problem is reduced to a system of two or three equations of the second degree, then Diophantus finds such rational expressions of the unknowns through one unknown and parameters under which all equations, except for one, turn into identities. From the remaining equation, he expresses the main unknown in terms of parameters, and then finds other unknowns as well.

The methods developed in Book II are applied by Diophantus to the more difficult problems of Book III, connected with systems of three, four, or more equations of degree not higher than two. He, in addition, before the formal solution of problems, conducts research and finds the conditions that the parameters must satisfy in order for solutions to exist.

In Book IV there are definite and indefinite equations of third and higher degrees. Here the situation is much more complicated, because, generally speaking, the unknowns cannot be expressed as rational functions of one parameter. But, as before, if one or two rational points of the cubic curve fz (x, y)== 0, then other points can be found. Diophantus uses new methods in solving the problems of Book IV.

Book V contains the most difficult tasks; some of them are solved using equations of the third and fourth degrees from three or more unknowns. There are also those in which it is required to decompose a given integer into the sum of two, three or four squares, and these squares must satisfy certain inequalities.,

When solving problems, Diophantus considers the Pell equation twice ax 2 + 1 = at 2 .

The problems in Book VI concern right-angled triangles with rational sides. To the condition X 2 + at 2 == z 2 they add more conditions regarding areas, perimeters, sides of triangles.

Book VI proves that if the equation ax 2 + b == at 2 has at least one rational solution, then there will be an infinite number of them. To solve the problems of book VI, Diophantus uses all the methods he uses.

By the way, in one of the ancient handwritten collections of tasks in verse, the life of Diophantus is described in the form of the following algebraic riddle, representing the tombstone inscription on his grave

The ashes of Diophantus the tomb rests; marvel at her - and a stone

The age of the departed will tell him with wise art.

By the will of the gods, he lived a sixth of his life as a child.

And he met half of the sixth with fluff on his cheeks.

Only the seventh passed, he got engaged to his girlfriend.

After spending five years with her, the wise man waited for his son;

His beloved son lived only half his father's life.

He was taken from his father by his early grave.

Twice two years the parent mourned the heavy grief,

Here I saw the limit of my sad life.

The riddle task is reduced to compiling and solving the equation:

whence x = 84 = that's how many years Diophantus lived.

Indefinite equation x 2 + y 2 = z 2

Introduction

It can be seen that over a period of more than one and a half thousand years, mathematical science in Greece had significant achievements.

In the history of mathematics, the period of existence of the Alexandrian school we have considered is called the "First Alexandrian school". From the beginning of our era, on the basis of the works of the Alexandrian mathematicians, the rapid development of idealistic philosophy begins: the ideas of Plato and Pythagoras are revived again, and this philosophy of the Neoplatonists and Neopythagoreans quickly reduces the scientific significance of the works of new representatives of mathematical thought. But the weight of mathematical thought does not fade, and from time to time it appears in the works of individual mathematicians, such as Diophantus.

The development of algebra was hampered by the fact that symbolic notation had not yet come into use sufficiently, a hint of which we first encounter in the works of Diophantus, who used only individual symbols and abbreviations of notation.

The purpose of the work is to study the arithmetic of Diophantus.

Biography of Diophantus

Diophantus presents one of the most difficult riddles in the history of science. We do not know either the time when he lived, or his predecessors who would have worked in the same area. His works are like a sparkling fire in the midst of complete impenetrable darkness.

The period of time when Diophantus could have lived is half a millennium! The lower bound of this interval is determined without difficulty: in his book on polygonal numbers, Diophantus repeatedly mentions the mathematician Hypsicles of Alexandria, who lived in the middle of the 2nd century BC. On the other hand, in the comments of Theon of Alexandria to the "Almagest" by the famous astronomer Ptolemy, an excerpt from the work of Diophantus is placed. Theon lived in the middle of the 4th century AD. This defines the upper bound of this interval. So, 500 years!

The French historian of science Paul Tannery, publisher of the most comprehensive text of Diophantus, has attempted to narrow this gap. In the library of Escurial, he found excerpts from a letter from Michael Psellos, a Byzantine scholar of the 11th century, which says that “the most learned Anatoly, after collecting the most essential parts of this science (we are talking about the introduction of degrees of the unknown and about their designations), dedicated them to his friend Diophantus." Anatoly of Alexandria really compiled an "Introduction to Arithmetic", excerpts from which are given in the works of Iamblichus and Eusebius that have come down to us. But Anatoly lived in Alexandria in the middle of the 3rd century AD. and even more precisely, until the year 270, when he became bishop of Laodacia. This means that his friendship with Diophantus, whom everyone calls the Alexandrian, must have taken place before that. So, if the famous Alexandrian mathematician and a friend of Anatoly named Diophantus are one person, then the life of Diophantus is the middle of the 3rd century AD.

The “Arithmetic” of Diophantus itself is dedicated to the “venerable Dionysius”, who, as can be seen from the text of the “Introduction”, was interested in arithmetic and its teaching. Although the name Dionysius was quite common at the time, Tannery suggested that the "venerable" Dionysius should be sought among famous people of the era who held prominent positions. And then it turned out that in 247 a certain Dionysius became the bishop of Alexandria, who from 231 led the city's Christian gymnasium! Therefore, Tannery identified this Dionysius with the one to whom Diophantus devoted his work, and came to the conclusion that Diophantus lived in the middle of the 3rd century AD. We may, for lack of a better way, accept this date.

But the place of residence of Diophantus is well known - this is the famous Alexandria, the center of scientific thought of the Hellenistic world.

After the collapse of the vast empire of Alexander the Great, Egypt at the end of the 4th century BC. went to his commander Ptolemy Lag, who moved the capital to a new city - Alexandria. Soon this multilingual trading city became one of the most beautiful cities of antiquity. Rome later surpassed it in size, but for a long time it was not equal. And it was this city that became for many centuries the scientific and cultural center of the ancient world. This was due to the fact that Ptolemy Lag founded the Museum, the temple of the Muses, something like the first Academy of Sciences, where the most prominent scientists were invited, and the content was assigned to them, so that their main business was reflection and conversation with students. A famous library was built at the Museuon, which in its best days contained more than 700,000 manuscripts. It is not surprising that scientists and young men thirsting for knowledge from all over the world rushed to Alexandria to listen to famous philosophers, learn astronomy and mathematics, and have the opportunity to delve into the study of unique manuscripts in the cool halls of the library.

The museum survived the Ptolemaic dynasty. In the first centuries BC. it fell into a temporary decline associated with the general decline of the Ptolemaic house in connection with the Roman conquests (Alexandria was finally conquered in 31 BC), but then in the first centuries AD. it revived again, already supported by the Roman emperors. Alexandria continued to be the scientific center of the world. Rome has never been its rival in this respect: Roman science (we mean the natural sciences) simply did not exist, and the Romans remained faithful to the precepts of Virgil, who wrote:

Thinner others will forge life-breathing bronze, -

I believe that - they will create living faces from marble,

More eloquent will be in the courts, the movements of the sky

With a cane they will draw with their own and calculate the stars of ascension,

You, Roman, know how to rule the peoples.

And if in the III-II centuries BC. The museum shone with the names of Euclid, Apollonius, Eratosthenes, Hipparchus, then in the I-III centuries AD. scientists such as Heron, Ptolemy and Diophantus worked here.

In order to exhaust everything known about the personality of Diophantus, we present a riddle poem that has come down to us:

The ashes of Diophantus the tomb rests; marvel at her - and a stone

The age of the departed will tell him with wise art.

By the will of the gods, he lived a sixth of his life as a child

And he met half of the sixth with fluff on his cheeks.

Only the seventh passed, he got engaged to his girlfriend.

After spending five years with her, the sage waited for her son;

His beloved son lived only half his father's life.

He was taken from his father by his early grave.

Twice two years the parent mourned the heavy grief,

Here I saw the limit of my sad life.

From this it is easy to calculate that Diophantus lived for 84 years. However, for this it is not at all necessary to master the art of Diophantus! It is enough to be able to solve an equation of the 1st degree with one unknown, and Egyptian scribes were able to do this as far back as 2 thousand years BC.

Diophantus of Alexandria(ancient Greek; lat. Diophantus) - an ancient Greek mathematician who presumably lived in the 3rd century AD. e. Often referred to as the "father of algebra". Author of "Arithmetic" - a book devoted to finding positive rational solutions to indefinite equations. Nowadays, "Diophantine equations" are usually understood as equations with integer coefficients, the solutions of which must be found among integers.

Diophantus was the first Greek mathematician who considered fractions on an equal footing with other numbers. Diophantus was also the first among ancient scientists to propose a developed mathematical symbolism, which made it possible to formulate his results in a fairly compact form.

A crater on the visible side of the Moon is named after Diophantus.

Biography

Almost nothing is known about the details of his life. On the one hand, Diophantus quotes Hypsicles (2nd century BC); on the other hand, Theon of Alexandria (about 350 AD) writes about Diophantus, from which it can be concluded that his life proceeded within the boundaries of this period. A possible specification of the time of Diophantus's life is based on the fact that his Arithmetic is dedicated to "the most venerable Dionysius". It is believed that this Dionysius is none other than Bishop Dionysius of Alexandria, who lived in the middle of the 3rd century. n. e.

The Palatine Anthology contains an epigram-task:

The ashes of Diophantus the tomb rests; marvel at her - and the stone will tell the age of the deceased with his wise art. By the will of the gods, he lived a sixth of his life as a child. And I met half of the sixth with fluff on my cheeks. Only the seventh passed, he got engaged to his girlfriend. With her, after spending five years, the wise man waited for his son; His beloved son lived only half his father's life. He was taken from his father by his early grave. Twice two years the parent mourned the heavy grief, Here he saw the limit of his sad life. (Translated by S. P. Bobrov)

It is equivalent to solving the following equation:

This equation gives x = 84 (\displaystyle x=84) , so Diophantus's age is 84 years. However, the accuracy of the information cannot be confirmed.

Arithmetic of Diophantus

The main work of Diophantus - Arithmetic in 13 books. Unfortunately, only 6 of the first 13 books have survived.

The first book is preceded by an extensive introduction, which describes the notation used by Diophantus. Diophantus calls the unknown "number" () and denotes it with a letter, the square of the unknown - with a symbol (short for - "degree"), the cube of the unknown - with a symbol (short for - "cube"). Special signs are provided for the next degrees of the unknown, up to the sixth, called the cubo-cube, and for their opposite degrees, up to the minus sixth.

Diophantus does not have an addition sign: he simply writes positive terms side by side in descending order of degree, and in each term the degree of the unknown is first written, and then the numerical coefficient. Subtracted members are also written side by side, and a special sign in the form of an inverted letter is placed in front of their entire group. The equal sign is indicated by two letters (short for "equal").

The rule of reduction of similar terms and the rule of adding or subtracting the same number or expression to both parts of the equation are formulated: what later al-Khwarizmi called "algebra and almukabala". A rule of signs has been introduced: “a minus by a plus gives a minus”, “a minus by a minus gives a plus”; this rule is used when multiplying two expressions with subtractive terms. All this is formulated in a general way, without reference to geometric interpretations.

Most of the work is a collection of problems with solutions (there are 189 of them in the surviving six books), skillfully chosen to illustrate general methods. The main problem of Arithmetic is finding positive rational solutions to indefinite equations. Rational numbers are treated by Diophantus in the same way as natural numbers, which is not typical for ancient mathematicians.

Biography

Latin translation Arithmetic (1621)

Almost nothing is known about the details of his life. On the one hand, Diophantus quotes Hypsicles (2nd century BC); on the other hand, Theon of Alexandria (about 350 AD) writes about Diophantus, from which it can be concluded that his life proceeded within the boundaries of this period. A possible specification of the time of Diophantus's life is based on the fact that his Arithmetic dedicated to "the most venerable Dionysius". It is believed that this Dionysius is none other than Bishop Dionysius of Alexandria, who lived in the middle of the 3rd century. n. e.

Arithmetic Diophantus

The main work of Diophantus - Arithmetic in 13 books. Unfortunately, only 6 of the first 13 books have survived.

The first book is preceded by an extensive introduction, which describes the notation used by Diophantus. Diophantus calls the unknown "number" ( ἀριθμός ) and denoted by the letter ς , the square of the unknown - a symbol (short for δύναμις - "degree"). Special signs are provided for the next degrees of the unknown, up to the sixth, called the cubo-cube, and for their opposite degrees. Diophantus does not have an addition sign: he simply writes positive terms next to each other, and in each term the degree of the unknown is first written, and then the numerical coefficient. The terms to be subtracted are also written side by side, and a special sign in the form of an inverted letter Ψ is placed in front of their entire group. The equal sign is indicated by two letters ἴσ (short for ἴσος - "equal"). The rule of reduction of similar terms and the rule of adding or subtracting the same number or expression to both parts of the equation are formulated: what later al-Khwarizmi called "algebra and almukabala". A rule of signs has been introduced: a minus times a minus gives a plus; this rule is used when multiplying two expressions with subtractive terms. All this is formulated in a general way, without reference to geometric interpretations.

Most of the work is a collection of problems with solutions (there are only 189 in the six books that have survived), skillfully chosen to illustrate general methods. Main issue Arithmetic- finding positive rational solutions to indefinite equations. Rational numbers are treated by Diophantus in the same way as natural numbers, which is not typical for ancient mathematicians.

First, Diophantus explores systems of 2nd order equations in 2 unknowns; it specifies a method for finding other solutions if one is already known. Then he applies similar methods to equations of higher degrees.

In the 10th century Arithmetic was translated into Arabic, after which the mathematicians of the countries of Islam (Abu Kamil and others) continued some studies of Diophantus. In Europe, interest in Arithmetic increased after Raphael Bombelli discovered this essay in the Vatican Library and published 143 problems from it in his Algebra(). In 1621, a classical, extensively commented Latin translation appeared. Arithmetic by Bacher de Meziriac. The methods of Diophantus were a huge influence on François Vieta and Pierre Fermat; however, in modern times, indefinite equations are usually solved in integers, and not in rational ones, as Diophantus did.

In the 20th century, under the name of Diophantus, an Arabic text of 4 more books was discovered Arithmetic. I. G. Bashmakova and E. I. Slavutin, after analyzing this text, put forward the hypothesis that their author was not Diophantus, but a commentator well versed in the methods of Diophantus, most likely Hypatia.

Other writings by Diophantus

Treatise of Diophantus About polygonal numbers (Περὶ πολυγώνων ἀριθμῶν ) is not completely preserved; in the surviving part, a number of auxiliary theorems are deduced by methods of geometric algebra.

From the writings of Diophantus About measuring surfaces (ἐπιπεδομετρικά ) and About multiplication (Περὶ πολλαπλασιασμοῦ ) also survived only fragments.

Book of Diophantus porisms known only from a few theorems used in Arithmetic.

Literature

Categories:

• Ancient Greek mathematicians
• Mathematicians of Ancient Rome
• Personalities in alphabetical order
• Mathematicians alphabetically
• 3rd century mathematicians
• Mathematicians in number theory

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See what "Diophantus of Alexandria" is in other dictionaries:

- (ca. 3rd century) ancient Greek mathematician. In the main work Arithmetic (6 books out of 13 have been preserved), he gave a solution to problems leading to the so-called. Diophantine equations, and for the first time introduced literal symbolism into algebra ... Big Encyclopedic Dictionary

- (about III century), ancient Greek mathematician. In the main work "Arithmetic" (6 books out of 13 have been preserved), he gave a solution to problems leading to the so-called Diophantine equations, and for the first time introduced letter symbolism into algebra. * * * DIOPHANTH… … encyclopedic Dictionary

- (probably c. 250 AD, although an earlier date is possible), an ancient Greek mathematician who worked in Alexandria, the author of the treatise Arithmetic in 13 books (6 survived), devoted mainly to the study of indefinite equations (the so-called ... … Collier Encyclopedia

Diophantus: Diophantus (commander) (2nd century BC). Diophantus of Alexandria (III century AD) ancient Greek mathematician ... Wikipedia

Diophantus- Alexandria (Greek Diophantos), approx. 250, other Greek. mathematician. In his main work "Arithmetic" (b. h. preserved) used the computational methods of the Egyptians and Babylonians. Researched the definition and uncertainty, tasks (especially linear and ... ... Dictionary of antiquity

- (born 325, mind 409 A.D.) famous Alexandrian mathematician. There is almost no information about his life; even the dates of his birth and death are not entirely certain. D. lived for 84 years, as can be seen from the epitaph, compiled in the form of the following ... ... Encyclopedic Dictionary F.A. Brockhaus and I.A. Efron

Diophantus- DIOFÁNT of Alexandria (c. 3rd century), other Greek. mathematician. In the main tr. Arithmetic (6 books out of 13 have been preserved) gave a solution to problems leading to the so-called. Diophantine Urnia, and for the first time introduced letter symbolism into algebra ... Biographical Dictionary

Octopuses have 8 legs, starfish have 5.

How many marine animals are in the aquarium if there are 39 limbs in total?

Diophantus of Alexandria was an ancient Greek mathematician who probably lived in the 3rd century AD.

Almost nothing is known about the details of his life. On the one hand, Diophantus quotes Hypsicles (2nd century BC); on the other hand, Theon of Alexandria (about 350 AD) writes about Diophantus, from which it can be concluded that his life proceeded within the boundaries of this period. A possible specification of the time of Diophantus's life is based on the fact that his "Arithmetic" is dedicated to "the most venerable Dionysius." It is believed that this Dionysius is none other than Bishop Dionysius of Alexandria, who lived in the middle of the 3rd century. n. e.

The Palatine Anthology contains an epigram-task from which we can conclude that Diophantus lived for 84 years:

The ashes of Diophantus the tomb rests; marvel at her and the stone

The age of the departed will tell him with wise art.

By the will of the gods, he lived a sixth of his life as a child.

And he met half of the sixth with fluff on his cheeks.

Only the seventh passed, he got engaged to his girlfriend.

With her, after spending five years, the wise man waited for his son;

His beloved son lived only half his father's life.

He was taken from his father by his early grave.

Twice two years the parent mourned the heavy grief,

Here I saw the limit of my sad life.

Using modern methods of solving equations, you can calculate how many years Diophantus lived. Let's make and solve the equation:

The solution to this equation is the number 84. Thus, Diophantus lived for 84 years.

The main work of Diophantus is "Arithmetic" in 13 books. Unfortunately, only 6 of the first 13 books have survived.

The first book is preceded by an extensive introduction, which describes the notation used by Diophantus. Diophantus calls the unknown "number" (?ριθμ?ς) and designates it with the letter ς, the square of the unknown - with a symbol (short for δ?ναμις - "degree"). Special signs are provided for the next degrees of the unknown, up to the sixth, called the cubo-cube, and for their opposite degrees. Diophantus does not have an addition sign: he simply writes positive terms next to each other, and in each term the degree of the unknown is first written, and then the numerical coefficient. Subtracted members are also written side by side, and a special sign in the form of an inverted letter Ψ is placed in front of their entire group. The equal sign is denoted by two letters ?σ (short for ?σος - "equal"). The rule of reduction of similar terms and the rule of adding or subtracting the same number or expression to both parts of the equation are formulated: what later al-Khwarizmi called “al-jabr and al-muqabala”. A rule of signs has been introduced: a minus times a minus gives a plus; this rule is used when multiplying two expressions with subtractive terms. All this is formulated in a general way, without reference to geometric interpretations.

Most of the work is a collection of problems with solutions (there are 189 of them in the surviving six books), skillfully chosen to illustrate general methods. The main problem of "Arithmetic" is finding positive rational solutions to indefinite equations. Rational numbers are treated by Diophantus in the same way as natural numbers, which is not typical for ancient mathematicians.

First, Diophantus explores systems of 2nd order equations in 2 unknowns; it specifies a method for finding other solutions if one is already known. Then he applies similar methods to equations of higher degrees.

In the 10th century Arithmetic was translated into Arabic, after which the mathematicians of the countries of Islam (Abu Kamil and others) continued some studies of Diophantus. In Europe, interest in Arithmetic increased after Raphael Bombelli discovered this work in the Vatican Library and published 143 problems from it in his Algebra (1572). In 1621, the classic, detailed Latin translation of the Arithmetic by Bacher de Meziriac appeared. The methods of Diophantus had an enormous influence on François Vieta and Pierre de Fermat; served as a starting point in the studies of Gauss and Euler. However, in modern times, indefinite equations are usually solved in integers, and not in rational ones, as Diophantus did.

In the 20th century, under the name of Diophantus, the Arabic text of 4 more books of Arithmetic was discovered. Some historians of mathematics, having analyzed this text, put forward the hypothesis that their author was not Diophantus, but a commentator well versed in the methods of Diophantus, most likely Hypatia.

Diophantus's treatise On Polygonal Numbers (Περ? πολυγ?νων ?ριθμ?ν) has not been fully preserved; in the surviving part, a number of auxiliary theorems are deduced by methods of geometric algebra.

Of the works of Diophantus "On the Measurement of Surfaces" (?πιπεδομετρικ?) and "On Multiplication" (Περ? πολλαπλασιασμο?), only fragments have also survived.

Diophantus' Porisms is only known from a few theorems used in Arithmetic.

Today an equation of the form

where P- an integer function (for example, a polynomial with integer coefficients), and the variables take integer values, are called in honor of the ancient Greek mathematician - Diophantine.

Probably the most famous Diophantine equation is

His solutions are Pythagorean triples: (3; 4; 5), (6; 8; 10), (5; 12; 13), (12; 35; 37)…

Proof of the unsolvability in integers of the Diophantine equation

at (Fermat's Last Theorem) was completed by English mathematician Andrew Wiles in 1994.

Another example of a Diophantine equation is the Pell equation

where parameter n is not an exact square.

Hilbert's tenth problem is one of 23 problems proposed by David Hilbert on August 8, 1900 at the II International Congress of Mathematicians. In Hilbert's report, the formulation of the tenth problem is the shortest of all:

Let a Diophantine equation be given with arbitrary unknowns and integer rational numerical coefficients. Indicate a method by which it is possible, after a finite number of operations, to determine whether this equation is solvable in integer rational numbers.

The proof of the algorithmic unsolvability of this problem took about twenty years and was completed by Yuri Matiyasevich in 1970.

Largely thanks to the activities of Pappus of Alexandria (3rd century), information about ancient scientists and their works has come down to us. After Apollonius (from the 2nd century BC), a decline began in ancient science. New deep ideas do not appear. In 146 BC. e. Rome captures Greece, and in 31 BC. e. - Alexandria. Against the background of general stagnation and decline, the gigantic figure of Diophantus of Alexandria, the last of the great ancient mathematicians, the "father of algebra", stands out sharply.

The following mathematical objects bear the name of Diophantus:

• diophantine analysis
• diophantine approximations
• diophantine equations