Gourt > Science > Math > Number Theory > Diophantine Equations (2)
In mathematics, a Diophantine equation is an indeterminate 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. The word Diophantine refers to the Hellenistic mathematician of the 3rd century CE, Diophantus of Alexandria, Egypt 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.
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 Brahmagupta in the 6th century and much later by Fermat.
- , where and : These are the Thue equations, and are, in general, solvable.
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Bibliography on Hilbert's Tenth Problem - Searchable, ~400 items.
Meta Description: [ Bibliography on Hilbert's Tenth Problem. This bibliography is a part of the Computer Science Bibliography Collection. ]
Developing A General 2nd Degree Diophantine Equation x^2 + p = 2^n - Methods to solve these equations.
Diagonal Quartic Surfaces - Articles, computations and software in Magma and GP by Martin Bright.
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Diophantine Equations - Dave Rusin's guide to Diophantine equations.
Diophantine Geometry in Characteristic p - A survey by José Felipe Voloch.
Diophantine m-tuples - Sets with the property that the product of any two distinct elements is one less than a square. Notes and bibliography by Andrej Dujella.
Meta Description: [ Sets with the property that the
product of any two of its distinct elements is one less than a square ]
Diophantus Quadraticus - On-line Pell Equation solver by Michael Zuker.
Meta Description: [ RNA secondary structure, RNA folding,
iocomputnearest neighbor parameters, free energy minimization, bioinformatics,
molecular sequence analysis, computational biology, mfold, dynamic programming
]
Egyptian Fractions - Lots of information about Egyptian fractions collected by David Eppstein.
Hilbert's Tenth Problem - Statement of the problem in several languages, history of the problem, bibliography and links to related WWW sites.
Hilbert's Tenth Problem - Given a Diophantine equation with any number of unknowns and with rational integer coefficients: devise a process, which could determine by a finite number of operations whether the equation is solvable in rational integers.
Meta Description: [ What is Mathematics? Goedel's Theorem and Around. Textbook for students. Section 4. By K.Podnieks ]
Linear Diophantine Equations - A web tool for solving Diophantine equations of the form ax + by = c.
Pell's Equation - Record solutions.
Pythagorean Triples in JAVA - A JavaScript applet which reads a and gives integer solutions of a^2+b^2 = c^2.
Pythagorean Triplets - A Javascript calculator for pythagorean triplets.
Meta Description: [ Properties and Calculation of pythagorean Triples ]
Quadratic Diophantine Equation Solver - Dario Alpern's Java/JavaScript code that solves Diophantine equations of the form Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 in two selectable modes: solution only and step by step (or teach) mode. There is also a link to his description of the solving methods.
Meta Description: [ Solves quadratic Diophantine equations (integer equations of the form a x^2 + b xy + c y^2 + dx + ey + f = 0) ]
Rational and Integral Points on Higher-dimensional Varieties - Some of conjectures and open problems, compiled at AIM.
Rational Triangles - Triangles in the Euclidean plane such that all three sides are rational. With tables of Heronian and Pythagorean triples.
Solving General Pell Equations - John Robertson's treatise on how to solve Diophantine equations of the form x^2 - dy^2 = N.
The Erdos-Strauss Conjecture - The conjecture states that for any integer n 1 there are integers a, b, and c with 4/n = 1/a + 1/b + 1/c, a 0, b 0, c 0. The page establishes that the conjecture is true for all integers n, 1 n = 10^14. Tables and software by Allan Swett.
Thue Equations - Definition of the problem and a list of special cases that have been solved, by Clemens Heuberger.
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