
Review of Short Phrases and Links 
This Review contains major "Diophantine Equations" related terms, short phrases and links grouped together in the form of Encyclopedia article.
Definitions
 Diophantine equations are equations with integer coefficients and integer unknowns.
 Diophantine equations are different.
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 Diophantine equations are equations of polynomial expressions for which rational or integer solutions are sought.
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 Diophantine equations are named after the Greek mathematician Diophantus of Alexandria, whose book Arithmetica included a study of such equations.
 Diophantine equations are equations for which we want to know the integer solutions.
 Dave Rusin's guide to Diophantine equations.
 We show an interesting connection between twoway deterministic finite automata with monotonic counters and quadratic Diophantine equations.
 R. TIJDEMAN, Exponential Diophantine equations, In Number Theory, Walter de Gruyter, (1998) 523539.
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 The depth of the study of general Diophantine equations is shown by the characterisation of Diophantine sets as recursively enumerable.
 Rational points on hyperelliptic curves, diophantine equations, Chabauty methods, computational number theory.
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 Nonlinear Diophantine equations (Pythagorean triples, Fermat’s Last Theorem, sums of squares and Pell’s equation).
 Integers, divisibility, prime numbers, unique factorization, congruences, quadratic reciprocity, Diophantine equations and arithmetic functions.
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 Divisibility, prime numbers, modular arithmetic, Chinese Remainder Theorem, Diophantine equations.
 A similar algorithm solves this problem for linear Diophantine equations with any other number of unknowns.
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 Divisibility theory in the integers, the theory of congruences, diophantine equations, Fermat's theorem and generalizations, and other topics.
 MAGMA program MAGMA code to solve Diophantine equations of the form F(x)=G(y), for which Runge's condition is satisfied. Created by Szabolcs Tengely.
 A web tool for solving Diophantine equations of the form ax + by = c.
 John Robertson's treatise on how to solve Diophantine equations of the form x^2 dy^2 = N.
 We show that this case and variations of it are equivalent to the solvability of some special classes of systems of quadratic Diophantine equations.
 Properties of integers, congruences, diophantine equations, algebraic number fields.
 Actually, this complexity depends on the complexity of computing the minimal solutions of diophantine equations.
 We present some recent results from our research on methods for finding the minimal solutions to linear Diophantine equations over the naturals.
 Introduction to number theory, treating divisibility, congruences, linear Diophantine equations and quadratic residues.
 Study of divisibility, primes, congruences, diophantine equations and quadratic residues.
 Divisibility, congruences, primitive roots, number theoretic functions, diophantine equations, continued fractions, quadratic residues.
 As the name suggests, many problems that we now call Diophantine equations are addressed in the Arithmetica of Diophantus.
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 Since Diophantus's time, number theorists have found solutions for many Diophantine equations and have established the unsolvability of many other equations.
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 M.L.Perez), Xiquan Publ. House, 2000; smarandache.djvu Nigel P. Smart, The Algorithmic Resolution of Diophantine Equations, Cambridge Univ.
 Integer solutions to Diophantine equations. Binary quadratic forms and the elementary theory of elliptic curves.
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 R.J. STROEKER, B.M.M. DE WEGER, Elliptic binomial Diophantine equations, Math.Comp. 68 (1999) 12571281.
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 DIOPHANTINE EQUATIONS AND POWER INTEGRAL BASES : NEW COMPUTATIONAL METHODS. Boston : Birkhäuser, c2002.
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 Text (gzipped PS).Algorithmic Solution of Diophantine Equations  Thomas Stoll, TU Graz, 2001.
 The equation was eventually solved by Euler in the early 18th century, who also solved a number of other Diophantine equations.
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 International conference on diophantine equations in honour of Professor T.N. Shorey on his 60th Birthday.
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 XPath 1.0 does not include an exponentiation operator; however, this is not a problem, since the exponents in Diophantine equations are whole numbers.
 Elementary number theory, divisibility, fundamental theorem of arithmetic, prime numbers, quadratic reciprocity, Diophantine equations.
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 Also, some of the most famous problems of number theory, such as Fermat's Last Theorem, are Diophantine equations posed by mathematicians living much later.
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 On the other hand, the very old and famous 10 th Hilbert problem of solving Diophantine equations has been proved undecidable by Matijasevic [9].
 The theory of Diophantine equations has even been shown to be undecidable (see Hilbert's tenth problem).
 T. J. Chou and G. E. Collins. Algorithms for the solution of systems of linear Diophantine equations. SIAM J. Comput., 11(4):687708, Nov. 1982.
 The Euclidean Algorithm and Linear Diophantine Equations Chapter 3.
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 In addition, the problem of solving linear Diophantine equations will also be addressed.
 Okay today we are going to look at solving linear Diophantine equations.
 Abstract: a lower bound for the complexity of solving linear Diophantine equations such as knapsack problems on an idealised computer.
 One example of a linear system of Diophantine equations merits special mention: given integers a,b,c,d we seek an integer x with x = a mod b and x = c mod d.
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 For solving the system of Diophantine equations L in the rule Dio we use one of the known algorithms, e.g.
 Diophantine Equations  Dave Rusin's guide to Diophantine equations. Diophantine Geometry in Characteristic p  A survey by Jos Felipe Voloch.
 Diophantine equations, multiplicative functions, distribution of primes.
 On the Psixyology of Diophantine Equations PhD thesis, Pieter Moree, Leiden, 1993.
 Diophantine equations, Diophantine approximation.
 The theory of Diophantine equations has even been shown to be undecidable (see Hilbert's tenth problem).
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