In mathematics, a Diophantine equation is a polynomial equation that only allows the variables to be integers. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ... In mathematics, one often (not quite always) distinguishes between an identity, which is an assertion that two expressions are equal regardless of the values of any variables that occur within them, and an equation, which may be true for only some (or none) of the values of any such variables. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...

The word Diophantine refers to the Greek mathematician of the third century A.D., 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. Diophantus of Alexandria - Διόφαντος ο Αλεξανδρεύς - (circa 200/214 – circa 284/298) was a Greek mathematician. ... Antiquity and modernity stand cheek-by-jowl in Egypts chief Mediterranean seaport Located on the Mediterranean Sea coast, Alexandria Αλεξάνδρεια (in Arabic, الإسكندرية, transliterated al-ʼIskandariyyah) is the chief seaport in Egypt, and that countrys second largest city, and the capital of the Al Iskandariyah governate. ... Algebra is a branch of mathematics which studies structure and quantity. ...

A linear Diophantine equation is an equation between two sums of monomials of degree zero or one. In mathematics, a monomial is a particular kind of polynomial, having just one term. ...

Examples of Diophantine equations

• ax + by = 1: See Bézout's identity; this is a linear Diophantine.
• xⁿ + yⁿ = zⁿ: 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 above equation exist.
• x² - n y² = 1: (Pell's equation) which is named, mistakenly, after the English mathematician John Pell. It was studied by Fermat.
• , where and : These are the Thue equations, and are, in general, solvable.

In number theory, Bézouts identity, named after Étienne Bézout, is a linear diophantine equation. ... 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 Fermat Fermats last theorem (sometimes abbreviated as FLT and also called Fermats great theorem) is one of the most famous theorems in the history of mathematics. ... 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. ... Pierre de Fermat Pierre de Fermat (August 17, 1601 – January 12, 1665) was a French lawyer at the Parlement of Toulouse, southern France, and a mathematician who is given credit for the development of modern calculus. ... Thue equations are Diophantine equations of the form , where and . ...

Diophantine analysis

Traditional 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?

Hilbert's tenth problem

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. In 1900, in recognition of their depth, 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. 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 discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ... 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 algebraic geometry techniques in this field, has continued to grow as a result; since treating arbitrary equations is a dead end, attention turns to equations also having 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. ... In mathematics, simultaneous equations are a set of equations where variables are shared. ... 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 recursively enumerable. 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, often less suggestively called recursion theory, a countable set S is called recursively enumerable, computably enumerable, semi-decidable or provable if There is an algorithm that, when given an input â€” typically an integer or a tuple of integers or a sequence of characters â€” eventually halts if it...

The field of Diophantine approximation deals with the cases of Diophantine inequalities: 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. ...

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 is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and finally proven in 2002 by Preda Mihăilescu. ...

External links

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