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In mathematics, a Diophantine equation is an equation between two polynomials with integer coefficients with any number of unknowns. A Diophantine problem is given as a Diophantine equation, whose solutions are the possible assignments of integers for the unknowns for which the equation is satisfied.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.
A linear Diophantine equation is an equation between two sums of monomials of degree zero or one.
1 Examples of Diophantine equations
- ax + by = 1: See Bézout's identity; this is a linear Diophantine.
- xn + yn = zn: For n = 2 there are 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 PellJohn Pell ( March 1, 1610 December 12, 1685), was an English mathematician. He was born at Southwick in Sussex, where his father was minister. He was educated at Steyning, and entered Trinity College, Cambridge, at the age of thirteen. During his universi. It was studied by FermatPierre de Fermat ( August 17, 1601 January 12, 1665) was a French lawyer at the Parliament of Toulouse and a mathematician who is given credit for the development of modern calculus. In particular, he is the precursor of differential calculus with his met.
- , where and : These are the Thue equations, and are, in general, solvable.
2 Diophantine analyisis
2.1 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?
2.2 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 problemsHilbert's problems are a list of 23 problems in mathematics put forth by German mathematician David Hilbert in the Paris conference of the International Congress of Mathematicians in 1900. The problems were all unsolved at the time, and several of them tu. In 1970, a novel result in mathematical logicMathematical 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. Although the layperson may think that mathematical log known as Matiyasevich's theoremTheorems Diophantine equations Matiyasevich's theorem proven in 1970 by Yuri Matiyasevich, implies that Hilbert's tenth problem is unsolvable. This problem is the challenge to find a general algorithm which can decide whether a given system of Diophantine settled the problem negatively: in general Diophantine problems are unsolvable. The point of view of Diophantine geometry, which is the application of algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. It can be seen as the study of solution sets of systems of algebraic equations . When there is more than o techniques in this field, has continued to grow as a result; since treating arbitrary equations is a dead end, attention turns to equations having a geometric meaning also.
