Diophantine equation

In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and An indeterminate equation, in Mathematics, is an equation for which there is an infinite set of solutions for example 2x = y is a simple indeterminate equation In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French 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. In Algebraic geometry, an algebraic curve is an Algebraic variety of dimension one In Mathematics, an algebraic surface is an Algebraic variety of dimension two In Mathematics, especially in Geometry and Group theory, a lattice in R n is a Discrete subgroup of

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. This article focuses on the cultural aspects of the Hellenistic age for the historical aspects see Hellenistic period. Diophantus of Alexandria ( Greek: b between 200 and 214 d between 284 and 298 AD sometimes called "the father of Algebra " a title some claim should Alexandria ( Egyptian Arabic: اسكندريه Eskendereyya; Standard Arabic: ar الإسكندرية Al-Iskandariyya; Ἀλεξάνδρεια This is a listing of common symbols found within all branches of the science of Mathematics. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. 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. In Mathematics, the word monomial means two different things in the context of Polynomials The first meaning is a product of powers of Variables

While individual equations present a kind of puzzle and have been considered throughout history, the formulation of general theories of Diophantine equations (beyond 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

1 Examples of Diophantine equations
2 Diophantine analysis
3 Linear Diophantine equations
4 Exponential Diophantine equations
5 External links

Examples of Diophantine equations

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

$ax+by=1\,$

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

$x^n+y^n=z^n\,$

For n = 2 there are infinitely many solutions (x,y,z), the Pythagorean triples. A Pythagorean triple consists of three positive Integers a, b, and c, such that a 2 + b 2 = For larger values of n, Fermat's last theorem states that no positive integer solutions x, y, z satisfying the equation exist. Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like

$x^2-ny^2=\pm 1\,$

(Pell's equation) which is named after the English mathematician John Pell. Pell's equation is any Diophantine equation of the form x^2-ny^2=1\ where n is a nonsquare integer and x John Pell ( March 1, 1611 &ndash December 12, 1685) was an English Mathematician. It was studied by Brahmagupta in the 7th century, as well as by Fermat in the 17th century. Brahmagupta ( (598–668 was an Indian mathematician and astronomer. Pierre de Fermat pjɛːʁ dəfɛʁ'ma ( 17 August 1601 or 1607/8 &ndash 12 January 1665) was a French Lawyer at the

$\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$

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 Erdős–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

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?
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.

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. India, officially the Republic of India (भारत गणराज्य inc-Latn Bhārat Gaṇarājya; see also other Indian languages) is a country The Shulba Sutras or Śulbasūtras ( Sanskrit śulba: "string cord rope" are Sutra texts belonging to the Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. 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. Baudhāyana, (fl ca 800 BCE was an Indian mathematician whowas most likely also a priest Apastamba (circa 600 BC) attempts simultaneous Diophantine equations with up to five unknowns.

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). Āryabhaṭa ( Devanāgarī: आर्यभट (AD 476 &ndash 550 is the first in the line of great mathematician-astronomers from the classical age of Indian mathematics Events By place Asia Kavadh I of Persia deposes his brother Djamasp and restores himself as king of Persia. 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.

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. Bhāskara (commonly called Bhāskara I to avoid confusion with the 12th century mathematician Bhāskara II) (c 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.

Brahmagupta (628) handled more difficult Diophantine equations - he investigated 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. Brahmagupta ( (598–668 was an Indian mathematician and astronomer. Pell's equation is any Diophantine equation of the form x^2-ny^2=1\ where n is a nonsquare integer and x 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; its solution was found seventy years later by Euler. This article is about the country For a topic outline on this subject see List of basic France topics. Pierre de Fermat pjɛːʁ dəfɛʁ'ma ( 17 August 1601 or 1607/8 &ndash 12 January 1665) was a French Lawyer at the However, the solution to this equation had been recorded centuries earlier by Bhaskara II (1150) (who also found the solution to Pell's equation), using a modified version of Brahmagupta's method. Bhaskara (1114 &ndash 1185 also known as Bhaskara II and Bhaskara Achārya ("Bhaskara the teacher" was an Indian mathematician Pell's equation is any Diophantine equation of the form x^2-ny^2=1\ where n is a nonsquare integer and x

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. Pierre de Fermat pjɛːʁ dəfɛʁ'ma ( 17 August 1601 or 1607/8 &ndash 12 January 1665) was a French Lawyer at the Arithmetica is an ancient Greek text on Mathematics written by the Mathematician Diophantus in the 3rd century CE. " 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. In Mathematics, a conjecture is a Mathematical statement which appears resourceful but has not been formally proven to be true under the rules of Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like It wasn't until 1994 that it was proven by the British mathematician Andrew Wiles. The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom, the UK or Britain,is a Sovereign state located Sir Andrew John Wiles KBE FRS (born 11 April 1953 is a British Mathematician and a professor at Princeton University

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.

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. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most Hilbert's tenth problem is the tenth on the list of Hilbert's problems of 1900 Hilbert's problems are a list of twenty-three problems in Mathematics put forth by German Mathematician David Hilbert at the Paris In 1970, a novel result in mathematical logic known as Matiyasevich's theorem settled the problem negatively: in general Diophantine problems are unsolvable. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. In Mathematics, a Diophantine set of j - Tuples of Integers is a set S for which there is some Polynomial with integer

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. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with 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. In Mathematics simultaneous equations are a set of Equations containing multiple variables In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients

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 Infinite descent is the traditional method, and has been pushed a long way. 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 Mathematics, a Diophantine set of j - Tuples of Integers is a set S for which there is some Polynomial with integer In Computability theory, traditionally called Recursion theory, a set S of Natural numbers is called recursively enumerable, computably 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 deals with the cases of Diophantine inequalities. In Number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of Real numbers by Rational 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 most celebrated single question in the field, the conjecture known as Fermat's Last Theorem, was cleared up by Andrew Wiles. In Mathematics, a conjecture is a Mathematical statement which appears resourceful but has not been formally proven to be true under the rules of Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like Sir Andrew John Wiles KBE FRS (born 11 April 1953 is a British Mathematician and a professor at Princeton University His journey through this proof can be found here: [1]. Other major results, such as Faltings' theorem, have disposed of old conjectures. In Number theory, the Mordell conjecture stated a basic result regarding the rational number solutions to Diophantine equations It was eventually proved by Gerd

Linear Diophantine equations

For more details on this topic, see Bézout's identity. In Number theory, Bézout's identity or Bézout's lemma is a linear Diophantine equation.

Linear Diophantine equations take the form of $a x + b y = c$ . 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. In Mathematics, the greatest common divisor (gcd, sometimes known as the greatest common factor (gcf or highest common factor (hcf, of two non-zero These can be found by applying the extended Euclidean algorithm. The extended Euclidean algorithm is an extension to the Euclidean algorithm for finding the Greatest common divisor (GCD of integers a and b 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 $a x + b y = c$ has no solutions.

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, however the majority are solved via Trial and Error. Catalan's conjecture (occasionally now referred to as Mihăilescu's theorem) is a Theorem in Number theory that was conjectured by the mathematician

External links

Diophantine Equation. From MathWorld at Wolfram Research. MathWorld is an online Mathematics reference work created and largely written by Eric W
Diophantine Equation. From PlanetMath. PlanetMath is a free, collaborative online Mathematics Encyclopedia.

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