81. Diophantine Equation From FOLDOC Online Computing Dictionary. Register a Domain. diophantine equation. mathematics Equations with integer coefficients to which integer solutions are sought. http://www.instantweb.com/foldoc/foldoc.cgi?Diophantine equation

82. DIOPHANTINE EQUATIONS DUE TO SMARANDACHE The diophantine equation y = 2x 1 x 2 x k +1 has infinitely many solutionsin distinct primes y, x 1 , x 2 , , x k . References 1 Ibstedt, H http://www.gallup.unm.edu/~smarandache/Dioph-Eq.htm

DIOPHANTINE EQUATIONS DUE TO SMARANDACHE 1) Conjecture: Let k > be an integer. There is only a finite number of solutions in integers p, q, x, y, each greater than 1, to the equation x p - y q = k. For k = 1 this was conjectured by Cassels (1953) and proved by Tijdeman (1976). References: [1] Ibstedt, H., Surphing on the Ocean of Numbers - A Few Smarandache Notions and Similar Topics , Erhus University Press, Vail, 1997, pp. 59-69. [2] Smarandache, F., Only Problems, not Solutions! , Xiquan Publ. Hse., Phoenix, 1994, unsolved problem #20. 2) Conjecture: Let k >= 2 be a positive integer. The diophantine equation y = 2x x ... x k has infinitely many solutions in distinct primes y, x , x , ..., x k References: [1] Ibstedt, H., Surphing on the Ocean of Numbers - A Few Smarandache Notions and Similar Topics , Erhus University Press, Vail, 1997, pp. 59-69. [2] Smarandache, F., Only Problems, not Solutions! , Xiquan Publ. Hse., Phoenix, fourth edition, 1994, unsolved problem #11.

83. Wauu.DE: Science: Math: Number Theory: Diophantine Equations http//liinwww.ira.uka.de/bibliography/Math/Hilbert10.html. Developing A General2nd Degree diophantine equation x^2 + p = 2^n Methods to solve these equations. http://www.wauu.de/Science/Math/Number_Theory/Diophantine_Equations/

Home Science Math Number Theory : Diophantine Equations Search DMOZ-Verzeichnis: All Categories Categories Onlye Kategorien: Equal Sums of Like Powers Fermat's Last Theorem Links: Sets of numbers such that the product of any two is one less than a square. Diophantus found the rational set 1/16, 33/16, 17/4, 105/16: Fermat the integer set 1, 3, 8, 120. http://www.weburbia.com/pg/diophant.htm Bibliography on Hilbert's Tenth Problem Searchable, ~400 items. http://liinwww.ira.uka.de/bibliography/Math/Hilbert10.html Developing A General 2nd Degree Diophantine Equation x^2 + p = 2^n Methods to solve these equations. http://www.biochem.okstate.edu/OAS/OJAS/thiendo.htm Diophantine Equations Dave Rusin's guide to Diophantine equations. http://www.math.niu.edu/~rusin/papers/known-math/index/11DXX.html Diophantine Geometry in Characteristic p A survey by Jos© Felipe Voloch. http://www.ma.utexas.edu/users/voloch/surveylatex/surveylatex.html 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. http://www.math.hr/~duje/dtuples.html

84. Outline: DIOPHANTINE EQUATIONS By transposing we can arrange that all the coefficients in the diophantine equationare positive integers eg 2x 3 y 2 3y 3 = x becomes 2x 3 y 2 =3y 3 + x . http://guru.math.carleton.ca/4803Dioph24Mar04.htm

Outline: DIOPHANTINE EQUATIONS March 24, 2004 History Diophantus 275 AD Alexandria, one of the great scientists in history for his introduction of notation for unknowns and their powers to study equations, without using geometry. For example he proves that any prime of the form 4m+3 can be written as the sum of two squares. A radical departure from earlier Greek mathematics; but much of his work is lost, we hardly even know his dates except by wine prices in his problems compared to known chart of wine prices for Alexandria that century. It was in reading a copy of Diophantus' Arithmetica that Fermat was inspired to conjecture his famous Fermat's Last Theorem and to scribble a note in the margin that he had a proof but the margin was too small. Notation (incl mention of Dioph contrib) ........ HilbertÕs 10 th problem Diophantus was interested in solving algebraic equations, but since algebra in his time did not recognize irrational, negative and complex numbers, he rejected equations with such solutions. We have come to call equations in several variables with integer coefficients Diophantine equations and we attempt to solve them for solutions in integers.

85. RR-1616 : On Efficiently Characterizing Solutions Of Linear Diophantine Equation Translate this page logo inria. RR-1616 - On efficiently characterizing solutions of lineardiophantine equations and its application to data dependence analysis. http://www.inria.fr/rrrt/rr-1616.html

RR-1616 - On efficiently characterizing solutions of linear diophantine equations and its application to data dependence analysis Eisenbeis, Christine Temam, O. Wijshoff, H. Rapport de recherche de l'INRIA- Rennes Fichier PostScript / PostScript file (1003 Ko) Fichier PDF / PDF file (1015 Ko) Equipe : Equipe : CALCPAR - Abstract : In this paper we present severals sets of mathematical tools for characterizin- g the solutions of linear Diophantine equations. First, a number of methods are given for reducing the complexity of the computations. Thereafter, we introduce different techniques for determining the exact number of solutions of linear Diophantine equations. Finally, we present a method for extracting efficiently the solutions of such equations. For all these methods the main focus has been put on their applicability and efficiency for data dependence analysis.

86. AMCA: On The Diophantine Equation $p^x-q^y=c$ By Florian Luca On the diophantine equation $p^xq^y=c$ by Florian Luca UNAM. The diophantineequation p x q y =c in positive integers (p, q, x, y http://at.yorku.ca/cgi-bin/amca/cakl-14

Atlas Mathematical Conference Abstracts Conferences Abstracts Organizers ... About AMCA Journées Arithmétiques XXIII July 612, 2003 University of Graz and University of Technology of Graz Graz, Styria, Austria Organizers S. Frisch, A. Geroldinger, P. Grabner, F. Halter-Koch, C. Heuberger, G. Lettl, R. Tichy View Abstracts Conference Homepage On the diophantine equation $p^x-q^y=c$ by Florian Luca UNAM The diophantine equation p x -q y =c in positive integers (p, q, x, y, c) with p and q distinct primes has received considerable interest. What is usually of interest for this equation is the following question: Given p and q, find all the values of c for which the above equation has at least two distinct solutions (x, y). The existence of two solutions (x, y) for the above equation reduces to the existence of a nontrivial solution of the equation p x -q y =p x -q y in positive integers (x , y , x , y ), where by nontrivial we mean that (x , y ) =/= (x , y In my talk, I will present two results concerning equation (1). I will first show that if one of the primes p and q, say p, is fixed, then there exist only finitely many quintuples of positive integers (q, x , y , x , y ) with q =/= p and prime which give a nontrivial solution of (1). I will then show that the ABC-conjecture implies that there should exist only finitely many sixtuples of positive integers (p, q, x

87. Avoiding Slack Variables In The Solving Of Linear Diophantine Equations And Ineq linear Diophantine systems of equations, which is itself a generalization of thealgorithm of Fortenbacher for solving a single linear diophantine equation. http://www.lri.fr/~contejea/tcs97.html

Avoiding Slack Variables in the Solving of Linear Diophantine Equations and Inequations Farid Ajili and Evelyne Contejean In this paper, we present an algorithm for solving directly linear Diophantine systems of both equations and inequations. Here directly means without adding slack variables for encoding inequalities as equalities. This algorithm is an extension of the algorithm due to Contejean and Devie for solving linear Diophantine systems of equations, which is itself a generalization of the algorithm of Fortenbacher for solving a single linear Diophantine equation. All the nice properties of the algorithm of Contejean and Devie are still satisfied by the new algorithm: it is complete, i.e. provides a (finite) description of the set of solutions, it can be implemented with a bounded stack, and it admits an incremental version. All of these characteristics enable its easy integration in the CLP paradigm. full paper

88. Department Of Mathematics And Statistics - Preprint (1998), Mollin, Richard. All Solutions of the diophantine equation X(2)-Dy(2)=n.Pre-print (1998), Publication Type Preprint. MR 800. Close Item. http://www.math.ucalgary.ca/research/preprint.php?pubid=216&__Goto=goto

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