![Rtriangle](https://upload.wikimedia.org/wikipedia/commons/6/6f/Rtriangle.svg)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Diophantine equation is an
equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
, typically a
polynomial equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation' ...
in two or more
unknown
Unknown or The Unknown may refer to:
Film
* The Unknown (1915 comedy film), ''The Unknown'' (1915 comedy film), a silent boxing film
* The Unknown (1915 drama film), ''The Unknown'' (1915 drama film)
* The Unknown (1927 film), ''The Unknown'' (1 ...
s with
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
coefficients, such that the only
solutions
Solution may refer to:
* Solution (chemistry), a mixture where one substance is dissolved in another
* Solution (equation), in mathematics
** Numerical solution, in numerical analysis, approximate solutions within specified error bounds
* Soluti ...
of interest are the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
ones. A linear Diophantine equation equates to a constant the sum of two or more
monomials
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer expone ...
, each of
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathematics
...
one. An exponential Diophantine equation is one in which unknowns can appear in
exponent
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
s.
Diophantine problems have fewer equations than unknowns and involve finding integers that solve simultaneously all equations. As such
systems of equations
In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single e ...
define
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
s,
algebraic surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...
s, or, more generally,
algebraic set
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a data ...
s, their study is a part of
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
that is called ''
Diophantine geometry
In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study ...
''.
The word ''Diophantine'' refers to the
Hellenistic mathematician of the 3rd century,
Diophantus
Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...
of
Alexandria
Alexandria ( or ; ar, ٱلْإِسْكَنْدَرِيَّةُ ; grc-gre, Αλεξάνδρεια, Alexándria) is the second largest city in Egypt, and the largest city on the Mediterranean coast. Founded in by Alexander the Great, Alexandria ...
, who made a study of such equations and was one of the first mathematicians to introduce
symbolism
Symbolism or symbolist may refer to:
Arts
* Symbolism (arts), a 19th-century movement rejecting Realism
** Symbolist movement in Romania, symbolist literature and visual arts in Romania during the late 19th and early 20th centuries
** Russian sy ...
into
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis.
While individual equations present 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.
Examples
In the following Diophantine equations, , , , and are the unknowns and the other letters are given constants:
Linear Diophantine equations
One equation
The simplest linear Diophantine equation takes the form , where , and are given integers. The solutions are described by the following theorem:
:''This Diophantine equation has a solution'' (where and are integers) ''if and only if'' ''is a multiple of the
greatest common divisor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...
of'' ''and'' . ''Moreover, if'' ''is a solution, then the other solutions have the form'' , ''where'' ''is an arbitrary integer, and'' ''and'' ''are the quotients of'' ''and'' ''(respectively) by the greatest common divisor of'' ''and'' .
Proof: If is this greatest common divisor,
Bézout's identity
In mathematics, Bézout's identity (also called Bézout's lemma), named after Étienne Bézout, is the following theorem:
Here the greatest common divisor of and is taken to be . The integers and are called Bézout coefficients for ; they ...
asserts the existence of integers and such that . If is a multiple of , then for some integer , and is a solution. On the other hand, for every pair of integers and , the greatest common divisor of and divides . Thus, if the equation has a solution, then must be a multiple of . If and , then for every solution , we have
:,
showing that is another solution. Finally, given two solutions such that , one deduces that . As and are
coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
,
Euclid's lemma
In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers, namely:
For example, if , , , then , and since this is divisible by 19, the lemma implies that one or both of 133 or 143 must be as w ...
shows that divides , and thus that there exists an integer such that and . Therefore, and , which completes the proof.
Chinese remainder theorem
The
Chinese remainder theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
describes an important class of linear Diophantine systems of equations: let be
pairwise coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
integers greater than one, be arbitrary integers, and be the product . The Chinese remainder theorem asserts that the following linear Diophantine system has exactly one solution such that , and that the other solutions are obtained by adding to a multiple of :
:
System of linear Diophantine equations
More generally, every system of linear Diophantine equations may be solved by computing the
Smith normal form In mathematics, the Smith normal form (sometimes abbreviated SNF) is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID). The Smith normal form of a matrix is diagonal, and can b ...
of its matrix, in a way that is similar to the use of the
reduced row echelon form
In linear algebra, a matrix is in echelon form if it has the shape resulting from a Gaussian elimination.
A matrix being in row echelon form means that Gaussian elimination has operated on the rows, and
column echelon form means that Gaussian el ...
to solve a
system of linear equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variable (math), variables.
For example,
:\begin
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\fracy-z=0
\end
is a system of three ...
over a field. Using
matrix notation
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object.
For example,
\begin ...
every system of linear Diophantine equations may be written
:,
where is an matrix of integers, is an
column matrix of unknowns and is an column matrix of integers.
The computation of the Smith normal form of provides two
unimodular matrices (that is matrices that are invertible over the integers and have ±1 as determinant) and of respective dimensions and , such that the matrix
:
is such that is not zero for not greater than some integer , and all the other entries are zero. The system to be solved may thus be rewritten as
:.
Calling the entries of and those of , this leads to the system
: for ,
: for .
This system is equivalent to the given one in the following sense: A column matrix of integers is a solution of the given system if and only if for some column matrix of integers such that .
It follows that the system has a solution if and only if divides for and for . If this condition is fulfilled, the solutions of the given system are
:
where are arbitrary integers.
Hermite normal form In linear algebra, the Hermite normal form is an analogue of reduced echelon form for matrices over the integers Z. Just as reduced echelon form can be used to solve problems about the solution to the linear system Ax=b where x is in R''n'', the H ...
may also be used for solving systems of linear Diophantine equations. However, Hermite normal form does not directly provide 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 is substantially easier to compute than the Smith normal form."
Integer linear programming
An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programming (ILP), in which the objective ...
amounts to finding some integer solutions (optimal in some sense) of linear systems that include also
inequation
In mathematics, an inequation is a statement that an inequality holds between two values. It is usually written in the form of a pair of expressions denoting the values in question, with a relational sign between them indicating the specific in ...
s. 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.
Homogeneous equations
A homogeneous Diophantine equation is a Diophantine equation that is defined by a
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; t ...
. A typical such equation is the equation of
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been k ...
:
As a homogeneous polynomial in indeterminates defines a
hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean ...
in the
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
of dimension , solving a homogeneous Diophantine equation is the same as finding the
rational point
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field ...
s of a projective hypersurface.
Solving a homogeneous Diophantine equation 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
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
is the th power of another rational number). A witness of the difficulty of the problem is Fermat's Last Theorem (for , 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 the number of solutions is finite (for example
Falting's theorem
In arithmetic geometry, the Mordell conjecture is the conjecture made by Louis Mordell that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points. In 1983 it was proved by Gerd Faltings, and ...
).
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.
Degree two
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 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 . For example, the Diophantine equation
:
does not have any other solution than the trivial solution . In fact, by dividing and by their
greatest common divisor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...
, one may suppose that they are
coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
. The squares modulo 4 are congruent to 0 and 1. Thus 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 and are all even, and are thus not coprime. Thus the only solution is the trivial solution . This shows that there is no
rational point
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field ...
on a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
of radius
centered at the origin.
More generally, the
Hasse principle In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of eac ...
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.
Geometric interpretation
Let
:
be a homogeneous Diophantine equation, where
is a
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
(that is, a homogeneous polynomial of degree 2), with integer coefficients. The ''trivial solution'' is the solution where all
are zero. If
is a non-trivial integer solution of this equation, then
are the
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
of a
rational point
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field ...
of the hypersurface defined by . Conversely, if
are homogeneous coordinates of a rational point of this hypersurface, where
are integers, then
is an integer solution of the Diophantine equation. Moreover, the integer solutions that define a given rational point are all sequences of the form
:
where is any integer, and is the greatest common divisor of the
It follows that solving the Diophantine equation
is completely reduced to finding the rational points of the corresponding projective hypersurface.
Parameterization
Let now
be an integer solution of the equation
As is a polynomial of degree two, a line passing through 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 , and the rational points are the those that are obtained from rational lines, that is, those that correspond to rational values of the parameters.
More precisely, one may proceed as follows.
By permuting the indices, one may suppose, without loss of generality that
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 is if all
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
s are zero at , all lines passing through are contained in the hypersurface, and one has a
cone
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines con ...
. The change of variables
:
does not change the rational points, and transforms into a homogeneous polynomial in variables. In this case, the problem may thus be solved by applying the method to an equation with fewer variables.
If the polynomial is a product of linear polynomials (possibly with non-rational coefficients), then it defines two
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
s. The intersection of these hyperplanes is a rational
flat
Flat or flats may refer to:
Architecture
* Flat (housing), an apartment in the United Kingdom, Ireland, Australia and other Commonwealth countries
Arts and entertainment
* Flat (music), a symbol () which denotes a lower pitch
* Flat (soldier), ...
, and contains rational singular points. This case is thus a special instance of the preceding case.
In the general case, let consider the
parametric equation
In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric obj ...
of a line passing through :
:
Substituting this in , one gets a polynomial of degree two in
that is zero for
It is thus divisible by
. The quotient is linear in
and may be solved for expressing
as a quotient of two polynomials of degree at most two in
with integer coefficients:
:
Substituting this in the expressions for
one gets, for ,
:
where
are polynomials of degree at most two with integer coefficients.
Then, one can return to the homogeneous case. Let, for ,
:
be the
homogenization
Homogeneity is a sameness of constituent structure.
Homogeneity, homogeneous, or homogenization may also refer to:
In mathematics
*Transcendental law of homogeneity of Leibniz
* Homogeneous space for a Lie group G, or more general transformati ...
of
These quadratic polynomials with integer coefficients form a parameterization of the projective hypersurface defined by :
:
A point of the projective hypersurface defined by is rational if and only if it may be obtained from rational values of
As
are homogeneous polynomials, the point is not changed if all
are multiplied by the same rational number. Thus, one may suppose that
are
coprime integers
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
. It follows that the integer solutions of the Diophantine equation are exactly the sequences
where, for ,
:
where is an integer,
are coprime integers, and is the greatest common divisor of the integers
One could hope that the coprimality of the
could imply that . Unfortunately this is not the case, as shown in the next section.
Example of Pythagorean triples
The equation
:
is probably the first homogeneous Diophantine equation of degree two that has been studied. Its solutions are the
Pythagorean triple
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...
s. This is also the homogeneous equation of the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
. In this section, we show how the above method allows retrieving
Euclid's formula
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...
for generating Pythagorean triples.
For retrieving exactly Euclid's formula, we start from the solution , corresponding to the point of the unit circle. A line passing through this point may be parameterized by its slope:
:
Putting this in the circle equation
:
one gets
:
Dividing by , results in
:
which is easy to solve in :
:
It follows
:
Homogenizing as described above one gets all solutions as
:
where is any integer, and are coprime integers, and is the greatest common divisor of the three numerators. In fact, if and are both odd, and if one is odd and the other is even.
The ''primitive triples'' are the solutions where and .
This description of the solutions differs slightly from Euclid's formula because Euclid's formula considers only the solutions such that and are all positive, and does not distinguish between two triples that differ by the exchange of and ,
Diophantine analysis
Typical questions
The questions asked in Diophantine analysis include:
#Are there any solutions?
#Are there any solutions beyond some that are easily found by
inspection
An inspection is, most generally, an organized examination or formal evaluation exercise. In engineering activities inspection involves the measurements, tests, and gauges applied to certain characteristics in regard to an object or activity. ...
?
#Are there finitely or infinitely many solutions?
#Can all solutions be found in theory?
#Can one in practice compute a full list of solutions?
These traditional problems often lay unsolved for centuries, and mathematicians gradually came to understand their depth (in some cases), rather than treat them as puzzles.
Typical problem
The given information is that a father's age is 1 less than twice that of his son, and that the digits making up the father's age are reversed in the son's age (i.e. ). This leads to the equation , thus . Inspection gives the result , , and thus equals 73 years and equals 37 years. One may easily show that there is not any other solution with and positive integers less than 10.
Many well known puzzles in the field of
recreational mathematics
Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...
lead to diophantine equations. Examples include the
cannonball problem
In the mathematics of figurate numbers, the cannonball problem asks which numbers are both square and square pyramidal. The problem can be stated as: given a square arrangement of cannonballs, for what size squares can these cannonballs also be a ...
,
Archimedes's cattle problem
Archimedes's cattle problem (or the or ) is a problem in Diophantine analysis, the study of polynomial equations with integer solutions. Attributed to Archimedes, the problem involves computing the number of cattle in a herd of the sun god from ...
and
the monkey and the coconuts.
17th and 18th centuries
In 1637,
Pierre de Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
scribbled on the margin of his copy of ''
Arithmetica'': "It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers." Stated in more modern language, "The equation has no solutions for any higher than 2." Following this, he wrote: "I have discovered a truly marvelous proof of this proposition, which this margin is too narrow to contain." Such a proof eluded mathematicians for centuries, however, and as such his statement became famous as
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been k ...
. It was not until 1995 that it was proven by the British mathematician
Andrew Wiles
Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specializing in number theory. He is best known for proving Fermat's Last Theorem, for which he was awar ...
.
In 1657, Fermat attempted to solve the Diophantine equation (solved by
Brahmagupta
Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical trea ...
over 1000 years earlier). The equation was eventually solved by
Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
in the early 18th century, who also solved a number of other Diophantine equations. The smallest solution of this equation in positive integers is , (see
Chakravala method
The ''chakravala'' method ( sa, चक्रवाल विधि) is a cyclic algorithm to solve indeterminate quadratic equations, including Pell's equation. It is commonly attributed to Bhāskara II, (c. 1114 – 1185 CE)Hoiberg & Ramchandani ...
).
Hilbert's tenth problem
In 1900,
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
proposed the solvability of all Diophantine equations as
the tenth
Tony S. Daniel, is an American comic book writer and artist, known for his work on various books for DC Comics, including ''Teen Titans'', '' Flash: The Fastest Man Alive'', and '' Batman''and '' Deathstroke'' and '' Nocterra'' as well as many ot ...
of his
fundamental problems. In 1970,
Yuri Matiyasevich
Yuri Vladimirovich Matiyasevich, (russian: Ю́рий Влади́мирович Матиясе́вич; born 2 March 1947 in Leningrad) is a Russian mathematician and computer scientist. He is best known for his negative solution of Hilbert's t ...
solved it negatively, building on work of
Julia Robinson
Julia Hall Bowman Robinson (December 8, 1919July 30, 1985) was an American mathematician noted for her contributions to the fields of computability theory and computational complexity theory—most notably in decision problems. Her work on Hilbe ...
,
Martin Davis, and
Hilary Putnam
Hilary Whitehall Putnam (; July 31, 1926 – March 13, 2016) was an American philosopher, mathematician, and computer scientist, and a major figure in analytic philosophy in the second half of the 20th century. He made significant contributions ...
to prove that a general
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
for solving all Diophantine equations
cannot exist.
Diophantine geometry
Diophantine geometry
In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study ...
, which is the application of techniques from
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
in this field, has continued to grow as a result; since treating arbitrary equations is a dead end, attention turns to equations that also have a geometric meaning. The central idea of Diophantine geometry is that of a
rational point
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field ...
, namely a solution to a polynomial equation or a
system of polynomial equations
A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations where the are polynomials in several variables, say , over some field .
A ''solution'' of a polynomial system is a set of values for the ...
, which is a vector in a prescribed
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
, when is ''not''
algebraically closed
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
.
Modern research
One of the few general approaches is through the
Hasse principle In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of eac ...
.
Infinite descent In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold f ...
is the traditional method, and has been pushed a long way.
The depth of the study of general Diophantine equations is shown by the characterisation of
Diophantine set
In mathematics, a Diophantine equation is an equation of the form ''P''(''x''1, ..., ''x'j'', ''y''1, ..., ''y'k'') = 0 (usually abbreviated ''P''(', ') = 0) where ''P''(', ') is a polynomial with integer coefficients, where ''x''1, ..., '' ...
s as equivalently described as
recursively enumerable
In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if:
*There is an algorithm such that the ...
. In other words, the general problem of Diophantine analysis is blessed or cursed with universality, and in any case is not something that will be solved except by re-expressing it in other terms.
The field of
Diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.
The first problem was to know how well a real number can be approximated by r ...
deals with the cases of ''Diophantine inequalities''. Here variables are still supposed to be integral, but some coefficients may be irrational numbers, and the equality sign is replaced by upper and lower bounds.
The single most celebrated question in the field, the
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...
known as
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been k ...
, was
solved by Andrew Wiles,
[ using tools from algebraic geometry developed during the last century rather than within number theory where the conjecture was originally formulated. Other major results, such as ]Faltings's theorem
In arithmetic geometry, the Mordell conjecture is the conjecture made by Louis Mordell that a curve of Genus (mathematics), genus greater than 1 over the field Q of rational numbers has only finitely many rational points. In 1983 it was proved by ...
, have disposed of old conjectures.
Infinite Diophantine equations
An example of an infinite diophantine equation is:
:, which can be expressed as "How many ways can a given integer be written as the sum of a square plus twice a square plus thrice a square and so on?" The number of ways this can be done for each forms an integer sequence. Infinite Diophantine equations are related to theta functions
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum fiel ...
and infinite dimensional lattices. This equation always has a solution for any positive . Compare this to:
:,
which does not always have a solution for positive .
Exponential Diophantine equations
If a Diophantine equation has as an additional variable or variables occurring as exponents
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
, it is an exponential Diophantine equation. Examples include the Ramanujan–Nagell equation In mathematics, in the field of number theory, the Ramanujan–Nagell equation is an equation between a square number and a number that is seven less than a power of two. It is an example of an exponential Diophantine equation, an equation to be ...
, , and the equation of the Fermat–Catalan conjecture
In number theory, the Fermat–Catalan conjecture is a generalization of Fermat's Last Theorem and of Catalan's conjecture, hence the name. The conjecture states that the equation
has only finitely many solutions (''a'',''b'',''c'',''m'',''n'','' ...
and Beal's conjecture, with inequality restrictions on the exponents. A general theory for such equations is not available; particular cases such as Catalan's conjecture
Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was Conjecture, conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderborn University. The integers 2 ...
have been tackled. However, the majority are solved via ad hoc methods such as Størmer's theorem or even trial and error
Trial and error is a fundamental method of problem-solving characterized by repeated, varied attempts which are continued until success, or until the practicer stops trying.
According to W.H. Thorpe, the term was devised by C. Lloyd Morgan (1 ...
.
See also
* Kuṭṭaka
Kuṭṭaka is an algorithm for finding integer solutions of linear Diophantine equations. A linear Diophantine equation is an equation of the form ''ax'' + ''by'' = ''c'' where ''x'' and ''y'' are unknown quantities and ''a'', ''b'', and ''c'' ar ...
, Aryabhata
Aryabhata (ISO: ) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the ''Aryabhatiya'' (which ...
's algorithm for solving linear Diophantine equations in two unknowns
Notes
References
*
*
*
*
*
Further reading
*Bashmakova, Izabella G. "Diophante et Fermat", ''Revue d'Histoire des Sciences'' 19 (1966), pp. 289–306
*Bashmakova, Izabella G. ''Diophantus and Diophantine Equations
''Diophantus and Diophantine Equations'' is a book in the history of mathematics, on the history of Diophantine equations and their solution by Diophantus of Alexandria. It was originally written in Russian by Isabella Bashmakova, and published by ...
''. Moscow: Nauka 1972 n Russian German translation: ''Diophant und diophantische Gleichungen''. Birkhauser, Basel/ Stuttgart, 1974. English translation: ''Diophantus and Diophantine Equations''. Translated by Abe Shenitzer with the editorial assistance of Hardy Grant and updated by Joseph Silverman. The Dolciani Mathematical Expositions, 20. Mathematical Association of America, Washington, DC. 1997.
*Bashmakova, Izabella G.
Arithmetic of Algebraic Curves from Diophantus to Poincaré"
''Historia Mathematica'' 8 (1981), 393–416.
*Bashmakova, Izabella G., Slavutin, E. I. ''History of Diophantine Analysis from Diophantus to Fermat''. Moscow: Nauka 1984 n Russian
*Bashmakova, Izabella G. "Diophantine Equations and the Evolution of Algebra", ''American Mathematical Society Translations'' 147 (2), 1990, pp. 85–100. Translated by A. Shenitzer and H. Grant.
*
*Rashed, Roshdi, Houzel, Christian. ''Les Arithmétiques de Diophante : Lecture historique et mathématique'', Berlin, New York : Walter de Gruyter, 2013.
*Rashed, Roshdi, ''Histoire de l'analyse diophantienne classique : D'Abū Kāmil à Fermat'', Berlin, New York : Walter de Gruyter.
External links
Diophantine Equation
From MathWorld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Dig ...
at Wolfram Research
Wolfram Research, Inc. ( ) is an American multinational company that creates computational technology. Wolfram's flagship product is the technical computing program Wolfram Mathematica, first released on June 23, 1988. Other products include Wo ...
.
*
Dario Alpern's Online Calculator
Retrieved 18 March 2009
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