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51-483: Equations in the book are presently called Diophantine equations . The method for solving these equations is known as Diophantine analysis . Most of the Arithmetica problems lead to quadratic equations . In Book 3, Diophantus solves problems of finding values which make two linear expressions simultaneously into squares or cubes. In book 4, he finds rational powers between given numbers. He also noticed that numbers of

102-463: A 1 , … , a n ) {\displaystyle \left(a_{1},\ldots ,a_{n}\right)} are the homogeneous coordinates of a rational point of the hypersurface defined by Q . Conversely, if ( p 1 q , … , p n q ) {\textstyle \left({\frac {p_{1}}{q}},\ldots ,{\frac {p_{n}}{q}}\right)} are homogeneous coordinates of

153-421: A d ) 2 = ( a d + b c ) 2 + ( a c − b d ) 2 {\displaystyle {\begin{alignedat}{4}\left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)&=(ac+db)^{2}+(bc-ad)^{2}\\&=(ad+bc)^{2}+(ac-bd)^{2}\\\end{alignedat}}} Diophantine equation In mathematics , a Diophantine equation

204-411: A x + b y + k ( a v − b u ) = a x + b y + k ( u d v − v d u ) = a x + b y , {\displaystyle {\begin{aligned}a(x+kv)+b(y-ku)&=ax+by+k(av-bu)\\&=ax+by+k(udv-vdu)\\&=ax+by,\end{aligned}}} showing that ( x + kv, y − ku )

255-407: A x + b y = c , {\displaystyle ax+by=c,} where a , b and c are given integers. The solutions are described by the following theorem: Proof: If d is this greatest common divisor, Bézout's identity asserts the existence of integers e and f such that ae + bf = d . If c is a multiple of d , then c = dh for some integer h , and ( eh, fh )

306-400: A kind of puzzle and have been considered throughout history, the formulation of general theories of Diophantine equations (beyond the case of linear and quadratic equations) was an achievement of the twentieth century. In the following Diophantine equations, w, x, y , and z are the unknowns and the other letters are given constants: The simplest linear Diophantine equation takes the form

357-405: A problem by Algebra: 1) An unknown is named and an equation is set up 2) An equation is simplified to a standard form( al-jabr and al-muqābala in arabic) 3) Simplified equation is solved Diophantus does not give classification of equations in six types like Al-Khwarizmi in extant parts of Arithmetica. He does says that he would give solution to three terms equations later, so this part of work

408-458: A rational point of this hypersurface, where q , p 1 , … , p n {\displaystyle q,p_{1},\ldots ,p_{n}} are integers, then ( p 1 , … , p n ) {\displaystyle \left(p_{1},\ldots ,p_{n}\right)} is an integer solution of the Diophantine equation. Moreover,

459-424: Is a quadratic form (that is, a homogeneous polynomial of degree 2), with integer coefficients. The trivial solution is the solution where all x i {\displaystyle x_{i}} are zero. If ( a 1 , … , a n ) {\displaystyle (a_{1},\ldots ,a_{n})} is a non-trivial integer solution of this equation, then (

510-429: Is a Diophantine equation that is defined by a homogeneous polynomial . A typical such equation is the equation of Fermat's Last Theorem As a homogeneous polynomial in n indeterminates defines a hypersurface in the projective space of dimension n − 1 , solving a homogeneous Diophantine equation is the same as finding the rational points of a projective hypersurface. Solving a homogeneous Diophantine equation

561-508: Is a part of algebraic geometry that is called Diophantine geometry . The word Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria , who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra . The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis . While individual equations present

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612-423: Is a polynomial of degree two, a line passing through A crosses the hypersurface at a single other point, which is rational if and only if the line is rational (that is, if the line is defined by rational parameters). This allows parameterizing the hypersurface by the lines passing through A , and the rational points are those that are obtained from rational lines, that is, those that correspond to rational values of

663-444: Is a product of linear polynomials (possibly with non-rational coefficients), then it defines two hyperplanes . The intersection of these hyperplanes is a rational flat , and contains rational singular points. This case is thus a special instance of the preceding case. In the general case, consider the parametric equation of a line passing through R : Substituting this in q , one gets a polynomial of degree two in x 1 , that

714-407: Is a solution. On the other hand, for every pair of integers x and y , the greatest common divisor d of a and b divides ax + by . Thus, if the equation has a solution, then c must be a multiple of d . If a = ud and b = vd , then for every solution ( x, y ) , we have a ( x + k v ) + b ( y − k u ) =

765-742: Is aggregation of objects of different types with no operations present For example, the Laurent polynomial written as 6 1 4 x − 1 + 25 x 2 − 9 {\displaystyle 6{\tfrac {1}{4}}x^{-1}+25x^{2}-9} in modern notation is written by Diophantus as "6 4 ′ inverse Powers, 25 Powers lacking 9 units", or "a collection of 6 1 4 {\displaystyle 6{\tfrac {1}{4}}} object of one kind with 25 object of second kind which lack 9 objects of third kind with no operation present". Similar to medieval Arabic algebra Diophantus uses three stages to solution of

816-460: Is an m × 1 column matrix of integers. The computation of the Smith normal form of A provides two unimodular matrices (that is matrices that are invertible over the integers and have ±1 as determinant) U and V of respective dimensions m × m and n × n , such that the matrix B = [ b i , j ] = U A V {\displaystyle B=[b_{i,j}]=UAV}

867-621: Is an equation , typically a polynomial equation in two or more unknowns with integer coefficients, for which only integer solutions are of interest. A linear Diophantine equation equates to a constant the sum of two or more monomials , each of degree one. An exponential Diophantine equation is one in which unknowns can appear in exponents . Diophantine problems have fewer equations than unknowns and involve finding integers that solve simultaneously all equations. As such systems of equations define algebraic curves , algebraic surfaces , or, more generally, algebraic sets , their study

918-434: Is an integer, t 1 , … , t n − 1 {\displaystyle t_{1},\ldots ,t_{n-1}} are coprime integers, and d is the greatest common divisor of the n integers F i ( t 1 , … , t n − 1 ) . {\displaystyle F_{i}(t_{1},\ldots ,t_{n-1}).} One could hope that

969-1102: Is another solution. Finally, given two solutions such that a x 1 + b y 1 = a x 2 + b y 2 = c , {\displaystyle ax_{1}+by_{1}=ax_{2}+by_{2}=c,} one deduces that u ( x 2 − x 1 ) + v ( y 2 − y 1 ) = 0. {\displaystyle u(x_{2}-x_{1})+v(y_{2}-y_{1})=0.} As u and v are coprime , Euclid's lemma shows that v divides x 2 − x 1 , and thus that there exists an integer k such that both x 2 − x 1 = k v , y 2 − y 1 = − k u . {\displaystyle x_{2}-x_{1}=kv,\quad y_{2}-y_{1}=-ku.} Therefore, x 2 = x 1 + k v , y 2 = y 1 − k u , {\displaystyle x_{2}=x_{1}+kv,\quad y_{2}=y_{1}-ku,} which completes

1020-464: Is completely reduced to finding the rational points of the corresponding projective hypersurface. Let now A = ( a 1 , … , a n ) {\displaystyle A=\left(a_{1},\ldots ,a_{n}\right)} be an integer solution of the equation Q ( x 1 , … , x n ) = 0. {\displaystyle Q(x_{1},\ldots ,x_{n})=0.} As Q

1071-458: Is considered outdated by Jeffrey Oaks and Jean Christianidis. The problems were solved on dust-board using some notation, while in books solution were written in "rhetorical style". Arithmetica also makes use of the identities: ( a 2 + b 2 ) ( c 2 + d 2 ) = ( a c + d b ) 2 + ( b c −

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1122-402: Is equivalent to the given one in the following sense: A column matrix of integers x is a solution of the given system if and only if x = Vy for some column matrix of integers y such that By = D . It follows that the system has a solution if and only if b i,i divides d i for i ≤ k and d i = 0 for i > k . If this condition is fulfilled, the solutions of

1173-619: Is generally a very difficult problem, even in the simplest non-trivial case of three indeterminates (in the case of two indeterminates the problem is equivalent with testing if a rational number is the d th power of another rational number). A witness of the difficulty of the problem is Fermat's Last Theorem (for d > 2 , there is no integer solution of the above equation), which needed more than three centuries of mathematicians' efforts before being solved. For degrees higher than three, most known results are theorems asserting that there are no solutions (for example Fermat's Last Theorem) or that

1224-400: Is if all partial derivatives are zero at R , all lines passing through R are contained in the hypersurface, and one has a cone . The change of variables does not change the rational points, and transforms q into a homogeneous polynomial in n − 1 variables. In this case, the problem may thus be solved by applying the method to an equation with fewer variables. If the polynomial q

1275-440: Is no solution. When a solution has been found, all solutions are then deduced. For proving that there is no solution, one may reduce the equation modulo p . For example, the Diophantine equation does not have any other solution than the trivial solution (0, 0, 0) . In fact, by dividing x, y , and z by their greatest common divisor , one may suppose that they are coprime . The squares modulo 4 are congruent to 0 and 1. Thus

1326-1078: Is possibly just lost In Arithmetica , Diophantus is the first to use symbols for unknown numbers as well as abbreviations for powers of numbers, relationships, and operations; thus he used what is now known as syncopated algebra . The main difference between Diophantine syncopated algebra and modern algebraic notation is that the former lacked special symbols for operations, relations, and exponentials. So for example, what would be written in modern notation as x 3 − 2 x 2 + 10 x − 1 = 5 , {\displaystyle x^{3}-2x^{2}+10x-1=5,} which can be rewritten as ( x 3 1 + x 10 ) − ( x 2 2 + x 0 1 ) = x 0 5 , {\displaystyle \left({x^{3}}1+{x}10\right)-\left({x^{2}}2+{x^{0}}1\right)={x^{0}}5,} would be written in Diophantus's syncopated notation as where

1377-429: Is substantially easier to compute than the Smith normal form." Integer linear programming amounts to finding some integer solutions (optimal in some sense) of linear systems that include also inequations . Thus systems of linear Diophantine equations are basic in this context, and textbooks on integer programming usually have a treatment of systems of linear Diophantine equations. A homogeneous Diophantine equation

1428-794: Is such that b i,i is not zero for i not greater than some integer k , and all the other entries are zero. The system to be solved may thus be rewritten as B ( V − 1 X ) = U C . {\displaystyle B(V^{-1}X)=UC.} Calling y i the entries of V X and d i those of D = UC , this leads to the system b i , i y i = d i , 1 ≤ i ≤ k 0 y i = d i , k < i ≤ n . {\displaystyle {\begin{aligned}&b_{i,i}y_{i}=d_{i},\quad 1\leq i\leq k\\&0y_{i}=d_{i},\quad k<i\leq n.\end{aligned}}} This system

1479-415: Is zero for x 1 = r 1 . It is thus divisible by x 1 – r 1 . The quotient is linear in x 1 , and may be solved for expressing x 1 as a quotient of two polynomials of degree at most two in t 2 , … , t n − 1 , {\displaystyle t_{2},\ldots ,t_{n-1},} with integer coefficients: Substituting this in

1530-614: The Astan Quds Library in Meshed (Iran) in a copy from 1198 AD. It was not catalogued under the name of Diophantus (but under that of Qusta ibn Luqa ) because the librarian was apparently not able to read the main line of the cover page where Diophantus’s name appears in geometric Kufi calligraphy . Arithmetica became known to mathematicians in the Islamic world in the tenth century when Abu'l-Wefa translated it into Arabic. Diophantus

1581-465: The Smith normal form of its matrix, in a way that is similar to the use of the reduced row echelon form to solve a system of linear equations over a field. Using matrix notation every system of linear Diophantine equations may be written A X = C , {\displaystyle AX=C,} where A is an m × n matrix of integers, X is an n × 1 column matrix of unknowns and C

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1632-570: The coprimality of the t i , could imply that d = 1 . Unfortunately this is not the case, as shown in the next section. The equation is probably the first homogeneous Diophantine equation of degree two that has been studied. Its solutions are the Pythagorean triples . This is also the homogeneous equation of the unit circle . In this section, we show how the above method allows retrieving Euclid's formula for generating Pythagorean triples. For retrieving exactly Euclid's formula, we start from

1683-417: The expressions for x 2 , … , x n − 1 , {\displaystyle x_{2},\ldots ,x_{n-1},} one gets, for i = 1, …, n − 1 , where f 1 , … , f n {\displaystyle f_{1},\ldots ,f_{n}} are polynomials of degree at most two with integer coefficients. Then, one can return to

1734-416: The form 4 n + 3 {\displaystyle 4n+3} cannot be the sum of two squares. Diophantus also appears to know that every number can be written as the sum of four squares . If he did know this result (in the sense of having proved it as opposed to merely conjectured it), his doing so would be truly remarkable: even Fermat, who stated the result, failed to provide a proof of it and it

1785-650: The given system are V [ d 1 b 1 , 1 ⋮ d k b k , k h k + 1 ⋮ h n ] , {\displaystyle V\,{\begin{bmatrix}{\frac {d_{1}}{b_{1,1}}}\\\vdots \\{\frac {d_{k}}{b_{k,k}}}\\h_{k+1}\\\vdots \\h_{n}\end{bmatrix}}\,,} where h k +1 , …, h n are arbitrary integers. Hermite normal form may also be used for solving systems of linear Diophantine equations. However, Hermite normal form does not directly provide

1836-693: The homogeneous case. Let, for i = 1, …, n , be the homogenization of f i . {\displaystyle f_{i}.} These quadratic polynomials with integer coefficients form a parameterization of the projective hypersurface defined by Q : A point of the projective hypersurface defined by Q is rational if and only if it may be obtained from rational values of t 1 , … , t n − 1 . {\displaystyle t_{1},\ldots ,t_{n-1}.} As F 1 , … , F n {\displaystyle F_{1},\ldots ,F_{n}} are homogeneous polynomials,

1887-421: The integer solutions that define a given rational point are all sequences of the form where k is any integer, and d is the greatest common divisor of the p i . {\displaystyle p_{i}.} It follows that solving the Diophantine equation Q ( x 1 , … , x n ) = 0 {\displaystyle Q(x_{1},\ldots ,x_{n})=0}

1938-419: The left-hand side of the equation is congruent to 0, 1, or 2, and the right-hand side is congruent to 0 or 3. Thus the equality may be obtained only if x, y , and z are all even, and are thus not coprime. Thus the only solution is the trivial solution (0, 0, 0) . This shows that there is no rational point on a circle of radius 3 , {\displaystyle {\sqrt {3}},} centered at

1989-409: The method of algebra, which existed before him. Algebra was practiced and diffused orally by practitioners, with Diophantus picking up technique to solve problems in arithmetic. In modern algebra a Laurent polynomial is linear combination of some variables, raised to integer powers, which behaves under multiplication, addition, and subtraction. Algebra of Diophantus, similar to medieval arabic algebra

2040-475: The modern parentheses and plus are used then the above equation can be rewritten as: ( x 3 1 + x 10 ) − ( x 2 2 + x 0 1 ) = x 0 5 {\displaystyle \left({x^{3}}1+{x}10\right)-\left({x^{2}}2+{x^{0}}1\right)={x^{0}}5} However the distinction between "rhetorical algebra", "syncopated algebra" and "symbolic algebra"

2091-442: The number of solutions is finite (for example Falting's theorem ). For the degree three, there are general solving methods, which work on almost all equations that are encountered in practice, but no algorithm is known that works for every cubic equation. Homogeneous Diophantine equations of degree two are easier to solve. The standard solving method proceeds in two steps. One has first to find one solution, or to prove that there

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2142-524: The origin. More generally, the Hasse principle allows deciding whether a homogeneous Diophantine equation of degree two has an integer solution, and computing a solution if there exist. If a non-trivial integer solution is known, one may produce all other solutions in the following way. Let be a homogeneous Diophantine equation, where Q ( x 1 , … , x n ) {\displaystyle Q(x_{1},\ldots ,x_{n})}

2193-510: The other solutions are obtained by adding to x a multiple of N : x = a 1 + n 1 x 1 ⋮ x = a k + n k x k {\displaystyle {\begin{aligned}x&=a_{1}+n_{1}\,x_{1}\\&\;\;\vdots \\x&=a_{k}+n_{k}\,x_{k}\end{aligned}}} More generally, every system of linear Diophantine equations may be solved by computing

2244-402: The parameters. More precisely, one may proceed as follows. By permuting the indices, one may suppose, without loss of generality that a n ≠ 0. {\displaystyle a_{n}\neq 0.} Then one may pass to the affine case by considering the affine hypersurface defined by which has the rational point If this rational point is a singular point , that

2295-538: The point is not changed if all t i are multiplied by the same rational number. Thus, one may suppose that t 1 , … , t n − 1 {\displaystyle t_{1},\ldots ,t_{n-1}} are coprime integers . It follows that the integer solutions of the Diophantine equation are exactly the sequences ( x 1 , … , x n ) {\displaystyle (x_{1},\ldots ,x_{n})} where, for i = 1, ..., n , where k

2346-411: The product n 1 ⋯ n k . {\displaystyle n_{1}\cdots n_{k}.} The Chinese remainder theorem asserts that the following linear Diophantine system has exactly one solution ( x , x 1 , … , x k ) {\displaystyle (x,x_{1},\dots ,x_{k})} such that 0 ≤ x < N , and that

2397-446: The proof. The Chinese remainder theorem describes an important class of linear Diophantine systems of equations: let n 1 , … , n k {\displaystyle n_{1},\dots ,n_{k}} be k pairwise coprime integers greater than one, a 1 , … , a k {\displaystyle a_{1},\dots ,a_{k}} be k arbitrary integers, and N be

2448-466: The solutions; to get the solutions from the Hermite normal form, one has to successively solve several linear equations. Nevertheless, Richard Zippel wrote that the Smith normal form "is somewhat more than is actually needed to solve linear diophantine equations. Instead of reducing the equation to diagonal form, we only need to make it triangular, which is called the Hermite normal form. The Hermite normal form

2499-538: The symbols represent the following: Unlike in modern notation, the coefficients come after the variables and addition is represented by the juxtaposition of terms. A literal symbol-for-symbol translation of Diophantus's syncopated equation into a modern symbolic equation would be the following: x 3 1 x 10 − x 2 2 x 0 1 = x 0 5 {\displaystyle {x^{3}}1{x}10-{x^{2}}2{x^{0}}1={x^{0}}5} where to clarify, if

2550-414: Was a Hellenistic mathematician who lived circa 250 AD, but the uncertainty of this date is so great that it may be off by more than a century. He is known for having written Arithmetica , a treatise that was originally thirteen books but of which only the first six have survived. Arithmetica is the earliest extant work present that solve arithmetic problems by algebra. Diophantus however did not invent

2601-639: Was not settled until Joseph Louis Lagrange proved it using results due to Leonhard Euler . Arithmetica was originally written in thirteen books, but the Greek manuscripts that survived to the present contain no more than six books. In 1968, Fuat Sezgin found four previously unknown books of Arithmetica at the shrine of Imam Rezā in the holy Islamic city of Mashhad in northeastern Iran. The four books are thought to have been translated from Greek to Arabic by Qusta ibn Luqa (820–912). Norbert Schappacher has written: [The four missing books] resurfaced around 1971 in

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