In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. In more technical language, they define an algebraic curve, algebraic surface or more general object, and ask about the lattice points on it. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... An indeterminate equation is an equation for which there is an infinite set of solutions – for example, 2x = y. ... In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ... The integers are commonly denoted by the above symbol. ... In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... In mathematics, an algebraic surface is an algebraic variety of dimension two. ... See lattice for other meanings of this term, both within and without mathematics. ...

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 Diophantus initiated is now called "Diophantine analysis". A linear Diophantine equation is an equation between two sums of monomials of degree zero or one. The term Hellenistic (derived from HéllÄ“n, the Greeks traditional self-described ethnic name) was established by the German historian Johann Gustav Droysen to refer to the spreading of Greek culture over the non-Greek people that were conquered by Alexander the Great. ... Title page of the 1621 edition of Diophantus Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac. ... Nickname: Alexandria on the map of Egypt Map of Alexandria Coordinates: , Country Egypt Founded 334 BC Government  - Governor Adel Labib Population (2001)  - City 3,500,000 Time zone EET (UTC+2)  - Summer (DST) EEST (UTC+3) Twin Cities  - Baltimore  United States  - Cleveland  United States  - ConstanÅ£a  Romania  - Durban  South Africa... Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ... In mathematics, a monomial is a particular kind of polynomial, having just one term. ...

While individual equations present a kind of puzzle and have been considered throughout history, the formulation of general theories of Diophantine equations (further to the theory of quadratic forms) was an achievement of the twentieth century. In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...

Examples of Diophantine equations

In the following Diophantine equations, x, y and z are the unknowns, the other letters being given.

 

This is a linear Diophantine equation (see the section "Linear Diophantine equations" below).

For n = 2 there are infinitely many solutions (x,y,z), the Pythagorean triples. For larger values of n, Fermat's last theorem states that no positive integer solutions x, y, z satisfying the equation exist.

 

(Pell's equation) which is named after the English mathematician John Pell. It was originally studied by Brahmagupta in the 6th century and much later by Fermat.

The Erdős–Straus conjecture states that, for every positive integer n ≥ 2, there exists a solution with x, y, and z all positive integers.

The Pythagorean theorem: a2 + b2 = c2 A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. ... Pierre de Fermats conjecture written in the margin of his copy of Arithmetica proved to be one of the most intriguing and enigmatic mathematical problems ever devised. ... Pells equation is any Diophantine equation of the form where n is a nonsquare integer. ... John Pell (March 1, 1610 - December 12, 1685), was an English mathematician. ... Brahmagupta (ब्रह्मगुप्त) (598-668) was an Indian mathematician and astronomer. ... Pierre de Fermat Pierre de Fermat IPA: (August 17, 1601–January 12, 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to modern calculus. ... The ErdÅ‘s–Straus conjecture states that for all integers n ≥ 2, the rational number 4/n can be expressed as the sum of three unit fractions. ...

Diophantine analysis

Traditional questions

The questions asked in Diophantine analysis include:

1. Are there any solutions?
2. Are there any solutions beyond some that are easily found by inspection?
3. Are there finitely or infinitely many solutions?
4. Can all solutions be found, in theory?
5. 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.

Diophantine analysis in India

India's contribution to integral solutions of Diophantine equations can be traced back to the Sulba Sutras, which were Indian mathematical texts written between 800 BC and 500 BC. Baudhayana (circa 800 BC) finds two sets of positive integral solutions to a set of simultaneous Diophantine equations, and also attempts simultaneous Diophantine equations with up to four unknowns. Apastamba (circa 600 BC) attempts simultaneous Diophantine equations with up to five unknowns. The Sulba Sutras or Sulva Sutras are a text of Vedic mathematics. ... This article is under construction. ... Baudhāyana, (fl. ... Apastamba (c. ...

Diophantine equations were later extensively studied by mathematicians in medieval India, who were the first to systematically investigate methods for determination of integral solutions of Diophantine equations. Systematic methods for finding integer solutions of Diophantine equations could be found in Indian texts from the time of Aryabhata AD (499). The first explicit description of the general integral solution of the linear Diophantine equation ay + bx = c occurs in his text Aryabhatiya. This algorithm is considered to be one of the most significant contributions of Aryabhata in pure mathematics. The technique was applied by Aryabhata to give integral solutions of simultaneous Diophantine equations of first degree, a problem with important applications in astronomy. Statue of Aryabhata on the grounds of IUCAA, Pune. ... Events March 1 - Pope Symmachus makes Antipope Laurentius bishop of Nocera in Campania. ...

Aryabhata describes the algorithm in just two stanzas of Aryabhatiya. His cryptic verses were elaborated by Bhaskara I (6th century) in his commentary Aryabhatiya Bhasya. Bhaskara I illustrated Aryabhata's rule with several examples including 24 concrete problems from astronomy. Without the explanation of Bhaskara I, it would have been difficult to interpret Aryabhata's verses. Bhaskara I aptly called the method kuttaka (pulverisation). The idea in kuttaka was later considered so important by the Indians that initially the whole subject of algebra used to be called kuttaka-ganita, or simply kuttaka. Bhāskara, or Bhāskara I, (c. ...

Brahmagupta (628) handled more difficult Diophantine equations - he discovered Pell's equation, and in his Samasabhavana he laid out a procedure to solve Diophantine equations of the second order, such as 61x² + 1 = y². These methods were unknown in the west, and this very equation was posed as a problem in 1657 by the French mathematician Pierre de Fermat - however, its solution was found only seventy years later by Euler. Meanwhile, many centuries ago, the solution to this equation was recorded by Bhaskara II (1150), using a modified version of Brahmagupta's method, and also found the solution to Pell's equation. Brahmagupta (ब्रह्मगुप्त) (598-668) was an Indian mathematician and astronomer. ... Pells equation is any Diophantine equation of the form where n is a nonsquare integer. ... Pierre de Fermat Pierre de Fermat IPA: (August 17, 1601–January 12, 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to modern calculus. ... Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ... Bhāskara (1114-1185), also called Bhāskara II and Bhāskarācārya (Bhaskara the teacher) was an Indian mathematician. ... Pells equation is any Diophantine equation of the form where n is a nonsquare integer. ...

17th and 18th centuries

In 1637, Pierre de Fermat 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 aⁿ + bⁿ = cⁿ has no solutions for any n higher than two." And then he wrote, intriguingly: "I have discovered a truly marvelous proof of this, which, however, the margin is not large enough to contain." Such a proof eluded mathematicians for centuries, however. As an unproven conjecture that eluded brilliant mathematicians' attempts to either prove it or disprove it for generations, his statement became famous as Fermat's last theorem. It wasn't until 1994 that it was proven by the British mathematician Andrew Wiles. Pierre de Fermat Pierre de Fermat IPA: (August 17, 1601–January 12, 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to modern calculus. ... Arithmetica, an ancient text on mathematics written by classical period Greek mathematician Diophantus in the second century AD is a collection of 130 algebra problems giving numerical solutions of determinate equations (those with a unique solution), and indeterminate equations. ... In mathematics, a conjecture is a mathematical statement which appears likely to be true, but has not been formally proven to be true under the rules of mathematical logic. ... Pierre de Fermats conjecture written in the margin of his copy of Arithmetica proved to be one of the most intriguing and enigmatic mathematical problems ever devised. ... For the French mathematician with work in the area of elliptic curves, see André Weil. ...

In 1657, Fermat attempted the Diophantine equation 61x² + 1 = y² (solved by Brahmagupta over 1000 years earlier). The equation was eventually solved by Euler in the early 18th century, who also solved a number of other Diophantine equations. Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ...

Hilbert's tenth problem

In 1900, in recognition of their depth, David Hilbert proposed the solvability of all Diophantine problems as the tenth of his celebrated problems. In 1970, a novel result in mathematical logic known as Matiyasevich's theorem settled the problem negatively: in general Diophantine problems are unsolvable. David Hilbert (January 23, 1862, Königsberg, East Prussia – February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ... Hilberts problems are a list of twenty-three problems in mathematics put forth by German mathematician David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900. ... Mathematical logic is a subfield of mathematics that is concerned with formal systems in relation to the way that they encode intuitive concepts of mathematical objects such as sets and numbers, proofs, and computation. ... Matiyasevichs theorem, proven in 1970 by Yuri Matiyasevich, implies that Hilberts tenth problem is unsolvable. ...

The point of view of Diophantine geometry, which is the application of techniques from algebraic geometry 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, namely a solution to a polynomial equation or system of simultaneous equations, which is a vector in a prescribed field K, when K is not algebraically closed. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... This article or section is in need of attention from an expert on mathematics. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero in F. In that case, every such polynomial splits into linear factors. ...

Modern research

One of the few general approaches is through the Hasse principle. Infinite descent is the traditional method, and has been pushed a long way. In mathematics, Helmut Hasses local-global principle, also known as the Hasse principle, is the assertion that an equation can be solved over the rational numbers if and only if it can be solved over the real numbers and over the p-adic numbers for every prime p. ... In mathematics, a proof by infinite descent is a particular kind of proof by mathematical induction. ...

The depth of the study of general Diophantine equations is shown by the characterisation of Diophantine sets as equivalently described as recursively enumerable. 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. In mathematics, a set S of j-tuples of integers is Diophantine precisely if there is some polynomial with integer coefficients f(n1, ..., nj, x1, ..., xk) such that a tuple (n1, ..., nj) of integers is in S if and only if there exist some integers x1, ..., xk with f(n1... In computability theory, traditionally called recursion theory, a set S of natural numbers is called recursively enumerable, computably enumerable, semidecidable or provable if: There is an algorithm that, when given an input number, eventually halts if and only if the input is an element of S. Or, equivalently, There is...

The field of Diophantine approximation 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. In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers. ...

The most celebrated single question in the field, the conjecture known as Fermat's Last Theorem, was cleared up by Andrew Wiles. His journey through this proof can be found here: [1]. Other major results, such as Faltings' theorem, have disposed of old conjectures. In mathematics, a conjecture is a mathematical statement which appears likely to be true, but has not been formally proven to be true under the rules of mathematical logic. ... Pierre de Fermats conjecture written in the margin of his copy of Arithmetica proved to be one of the most intriguing and enigmatic mathematical problems ever devised. ... For the French mathematician with work in the area of elliptic curves, see André Weil. ... In number theory, the Mordell conjecture stated a basic result regarding the rational number solutions to Diophantine equations. ...

Linear Diophantine equations

For more details on this topic, see Bézout's identity.

Linear Diophantine equations take the form of ax + by = c. If c is the greatest common divisor (gcd) of a and b, this is a Bézout's identity, and the equation has infinitely many solutions. These can be found by applying the extended Euclidean algorithm. It follows that there are also infinitely many solutions if c is a multiple of the gcd of a and b. If c is not a multiple of the gcd of a and b, then the Diophantine equation ax + by = c has no solutions. In number theory, Bézouts identity, named after Étienne Bézout, is a linear diophantine equation. ... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ... The extended Euclidean algorithm is an extension to the Euclidean algorithm for finding the greatest common divisor (GCD) of integers a and b: it also finds the integers x and y in Bézouts identity (Typically either x or y is negative). ...

Exponential Diophantine equations

If a Diophantine equation has as an additional variable or variables some integer(s) occurring as exponents, it is an exponential Diophantine equation. Such equations do not have a general theory; particular cases such as Mihăilescu's theorem have been tackled. In mathematics, exponentiation is a process generalized from repeated multiplication, in much the same way that multiplication is a process generalized from repeated addition. ... Mihăilescus theorem (formerly Catalans conjecture) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proved in 2002 by Preda Mihăilescu. ...

External links

• Diophantine Equation. From MathWorld at Wolfram Research.
• Diophantine Equation. From PlanetMath.