21. Math Forum - Ask Dr. Math We have searched the Web for information about diophantine equations. A diophantine equation is one equation in at least two variables, say x and y, http://mathforum.org/library/drmath/view/54241.html

Associated Topics Dr. Math Home Search Dr. Math Diophantine Equations Date: 11/17/97 at 15:37:25 From: Robert Felgate Subject: Diaphantine equations Dear Dr. Math, We have been investigating Pythagoras' theorem, and our teacher has suggested we search the web for information about diaphantine equations. We have not been very successful and wonder if you would help us. Thank you. Robert and his Mum http://mathforum.org/dr.math/ Associated Topics High School Linear Equations High School Number Theory Search the Dr. Math Library: Find items containing (put spaces between keywords): Click only once for faster results: [ Choose "whole words" when searching for a word like age. all keywords, in any order at least one, that exact phrase parts of words whole words Submit your own question to Dr. Math Math Forum Home Math Library Quick Reference ... Math Forum Search Ask Dr. Math TM http://mathforum.org/dr.math/

22. 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. http://www.math.leidenuniv.nl/~tengely/main2.html

23. Beal Conjecture C are coprime or, if not co-prime that any common factor could be divided out of each term until the equation existed with co-prime bases. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

24. PlanetMath: Diophantine Equation A diophantine equation is an equation for which the solutions are Generally,solving a diophantine equation is not as straightforward as solving a http://planetmath.org/encyclopedia/DiophantineEquation.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About Diophantine equation (Definition) A Diophantine equation is an equation for which the solutions are required to be integers Generally, solving a Diophantine equation is not as straightforward as solving a similar equation in the real numbers . For example, consider this equation: It is easy to find real numbers that satisfy this equation: pick any arbitrary and , and you can compute a from them. But if we require that all be integers, it is no longer obvious at all how to find solutions. Even though raising an integer to an integer power yields another integer, the reverse is not true in general. As it turns out, of course, there are no solutions to the above Diophantine equation: it is a case of Fermat's last theorem At the Second International Congress of Mathematicians in 1900, David Hilbert presented several unsolved problems in mathematics that he believed held special importance. Hilbert's tenth problem was to find a general procedure for determining if Diophantine equations have solutions:

25. On A Generalized Fermat-Wiles Equation Steven Finch's essay on the diophantine equation of the form x^n + y^n = c.z^n. http://www.mathsoft.com/mathresources/problems/article/0,,2186,00.html

26. The Abc Conjecture Comput. 68, No.225, 395401 (1999). Coh Cohn, J. H. E. The diophantine equation $(a\sp n-1)(b\sp n-1)=x\sp 2$. Period. Math. Hungar. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

27. The Prime Glossary: Diophantus Now we call an equation to be solved in integers a diophantine equation.For example, Diophantus considered the equations. ax + by = c http://primes.utm.edu/glossary/page.php?sort=Diophantus

28. The Beal Conjecture $75,000 prized problem pertaining to the diophantine equation of the form A^x + B^y = C^z where A, B, C, x, y and z are positive integers and x, y and z are all greater than 2, then A, B and C must have a common factor. http://www.math.unt.edu/~mauldin/beal.html

THE BEAL CONJECTURE AND PRIZE BEAL'S CONJECTURE: If A x +B y = C z , where A, B, C, x, y and z are positive integers and x, y and z are all greater than 2, then A, B and C must have a common prime factor. THE BEAL PRIZE. The conjecture and prize was announced in the December 1997 issue of the Notices of the American Mathematical Society. Since that time Andy Beal has increased the amount of the prize for his conjecture. The prize is now this: $100,000 for either a proof or a counterexample of his conjecture. The prize money is being held by the American Mathematical Society until it is awarded. In the meantime the interest is being used to fund some AMS activities and the annual Erdos Memorial Lecture. CONDITIONS FOR WINNING THE PRIZE. The prize will be awarded by the prize committee appointed by the American Mathematical Society. The present committee members are Charles Fefferman, Ron Graham, and Dan Mauldin. The requirements for the award are that in the judgment of the committee, the solution has been recognized by the mathematics community. This includes that either a proof has been given and the result has appeared in a reputable refereed journal or a counterexample has been given and verified. PRELIMINARY RESULTS.

29. Beal's Conjecture A Search For Counterexamples Beal's Conjecture is this There are no positive integers x m y n z r satisfying the equation xm + yn = zr http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

30. Exponential Diophantine Equations - Cambridge University Press Home Catalogue Exponential diophantine equations This is the firstintegrated presentation of the theory of exponential diophantine equations. http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=0521268265

31. Analytic Methods For Diophantine Equations And Diophantine Inequalities - Cambri Cubic forms the padic problem; 19. Homogeneous equations of higher degree; 20.A Diophantine inequality; Bibliography; Index. http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=0521605830

32. Generation5 - Genetic Algorithm Example: Diophantine Equation Generation5 aims to be the most comprehensive Artificial Intelligence site onthe Internet. Communityorientated, Generation5 deals with all AI topics http://www.generation5.org/content/1999/gaexample.asp

Home Articles Reviews Interviews ... Contribute Login Login: Password: Keep me logged in. ( help Register Why register? Lost your password? ... MobES Expert System Friends and Affiliates Home Articles Genetic Algorithms > Beginner Genetic Algorithm Example: Diophantine Equation By Samuel Hsiung and James Matthews Printable Version Make sure that you have read the genetic algorithms essay before reading this example. You also must have a working knowledge of C++ and object-oriented programming to utilize the classes and code examples provided. Genetic Algorithm Example Let us consider a diophantine (only integer solutions) equation: a+2b+3c+4d=30 , where a,b,c,d are positive integers. Using a genetic algorithm, all that is needed is a little time to reach a solution (a,b,c,d) . Of course you could ask, why not just use a brute force method (plug in every possible value for a,b,c,d given the constraints < a,b,c,d = )? The architecture of GA systems allow for a solution to be reached quicker since "better" solutions have a better chance of surviving and procreating, as opposed to randomly throwing out solutions and seeing which ones work. Let's start from the beginning. First we will choose 5 random initial solution sets, with constraints

33. Generation5 - Diophantine Equation Solver Generation5 aims to be the most comprehensive Artificial Intelligence site onthe Internet. Communityorientated, Generation5 deals with all AI topics http://www.generation5.org/content/2000/diophantine_ga.asp

Home Articles Reviews Interviews ... Contribute Login Login: Password: Keep me logged in. ( help Register Why register? Lost your password? ... MobES Expert System Friends and Affiliates Home Articles Genetic Algorithms > Applications/Code Diophantine Equation Solver By James Matthews Printable Version This is a C++ program that solves a diophantine equation using genetic algorithms. This is the first program is a new set of programs popping up on Generation5 - case studies. Therefore this page is a huge break down of the code and what it does, how it does it, and how to use it. Without further ado, here is the download link: diophantine.zip This code is the accompanying code to the genetic algorithms example , so please read that if you don't know anything about GAs before attempting to look at this code. Trust me, Sam wrote the essay, and I learnt and coded the program straight from what I learnt from that essay - its great! Now for the code: CDiophantine Firstly the class header (note for formatting reasons, a lot of the documentation is taken out): Firstly you notice that there are two structures, the gene structure and the actual CDiophantine class. The gene structure is used to keep track of the different solution sets. The population generated is a population of genes. The gene structure keeps track of its own fitness and likelihood values itself. I also coded a small function to test for equality, this just made some other code a lot more concise. Now onto the functions.

34. Diophantine Equation: Definition And Much More From Answers.com diophantine equation n. An algebraic equation with two or more variables whosecoefficients are integers, studied to determine all integral solutions. http://www.answers.com/topic/diophantine-equation

showHide_TellMeAbout2('false'); Business Entertainment Games Health ... More... On this page: Dictionary Wikipedia Best of Web Mentioned In Or search: - The Web - Images - News - Blogs - Shopping Diophantine equation Dictionary Diophantine equation n. An algebraic equation with two or more variables whose coefficients are integers, studied to determine all integral solutions. [After Diophantus , third-century A.D. Greek mathematician.] Wikipedia Diophantine equation 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. 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.

35. Matiyasevich's Theorem: Information From Answers.com A typical system of diophantine equations looks like this. 3x2y 7y2z3 = 18- 7y2 + 8z2 = 0. The question is whether there exist integers x, http://www.answers.com/topic/matiyasevich-s-theorem

showHide_TellMeAbout2('false'); Business Entertainment Games Health ... More... On this page: Wikipedia Mentioned In Or search: - The Web - Images - News - Blogs - Shopping Matiyasevich's theorem Wikipedia Matiyasevich's theorem Matiyasevich's theorem , proven in 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 equations polynomials with integer coefficients) has a solution among the integers. David Hilbert posed the problem in his 1900 address to the International Congress of Mathematicians. A typical system of diophantine equations looks like this: x y y z y z The question is whether there exist integers x y and z which satisfy both equations simultaneously. It turns out that it is always equivalent to ask whether a single Diophantine equation with several variables has any solutions among the natural numbers . For instance, the above system is solvable over the integers if and only if the following equation is solvable over the natural numbers: x x y y y y z z y y z z Matiyasevich utilized an ingenious trick involving Fibonacci numbers in order to show that solutions to Diophantine equations may grow exponentially . Earlier work by Julia Robinson Martin Davis and Hilary Putnam had shown that this suffices to show that no general algorithm deciding the solvability of Diophantine equations can exist.

36. Diophantine Equation -- From MathWorld research.htmlQuantitative finiteness results for diophantine equations, PhDThesis On thediophantine equation G_n(x)=G_m(y) with Q(x,y)=0, joint work with A. Pethõ http://www.astro.virginia.edu/~eww6n/math/DiophantineEquation.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Number Theory Diophantine Equations Diophantine Equation A Diophantine equation is an equation in which only integer solutions are allowed. Hilbert's 10th problem asked if an algorithm existed for determining whether an arbitrary Diophantine equation has a solution. Such an algorithm does exist for the solution of first-order Diophantine equations. However, the impossibility of obtaining a general solution was proven by Yuri Matiyasevich in 1970 (Matiyasevich 1970, Davis 1973, Davis and Hersh 1973, Davis 1982, Matiyasevich 1993) by showing that the relation (where is the th Fibonacci number ) is Diophantine. More specifically, Matiyasevich showed that there is a polynomial in , and a number of other variables , ... having the property that iff there exist integers , ... such that

37. Clemens Heuberger - Thue Equations After Diophantus von Alexandrien such equations are called diophantine So the research interest in diophantine equations is to find classes of such http://finanz.math.tu-graz.ac.at/~cheub/thue.html

Clemens Heuberger - Thue equations Diophantine equations Since antiquity, many people try to solve equations over the integers, Pythagoras for instance described all integers being the sides of a rectangular triangle. After Diophantus von Alexandrien such equations are called diophantine equations . Since that time, many mathematicians worked on this topic, such as Fermat, Euler, Kummer, Siegel, and Wiles. Among his 23 Problems, Hilbert raised the question, whether there exists an algorithm to solve any given polynomial diophantine equation; the negative answer has been given by Matijasevic in 1970. So the research interest in diophantine equations is to find classes of such equations which can be solved. Thue equations In 1909, A. Thue considered a special family of equations F(X,Y) = m, where F is an irreducible homogeneous form of degree n at least 3 and m is a nonzero integer. This type of equations is called after him since then; he proved that such an equation only has a finite number of solutions. His proof, however, is not constructive, so it does not lead to an algorithm. Only with Baker's lower bounds for linear forms in logarithms of algebraic numbers (19661968), effective bounds for the solution of many diophantine equations can be given. Since that time, many bounds have been improved and algorithms have been developed to solve one single Thue-equation in reasonable time on a computer (see Bilu and Hanrot). Parametrized Thue equations In 1990, E. Thomas studied a parametrized family of cubic Thue equations: It turns out that there exist only a few "trivial" solutions for large values of the parameter.

38. Historical Notes: Diophantine Equations diophantine equations. If variables appear only linearly, then it is possible to (The same is not true of simultaneous quadratic diophantine equations, http://www.wolframscience.com/reference/notes/1164b

SOME HISTORICAL NOTES From: Stephen Wolfram, A New Kind of Science Notes for Chapter 12: The Principle of Computational Equivalence Section: Implications for Mathematics and Its Foundations Page Diophantine equations. If variables appear only linearly, then it is possible to use ExtendedGCD (see page 946) to find all solutions to any system of Diophantine equations - or to show that none exist. Particularly from the work of Carl Friedrich Gauss around 1800 there emerged a procedure to find solutions to any quadratic Diophantine equation in two variables - in effect by reduction to the Pell equation x^2==a y^2+1 (see page 947), and then computing ContinuedFraction[Sqrt[a]]. The minimal solutions can be large; the largest ones for successive coefficient sizes are given below. (With size s coefficients it is for example known that the solutions must always be less than (14 s)^(5s).). There is a fairly complete theory of homogeneous quadratic Diophantine equations with three variables, and on the basis of results from the early and mid-1900s a finite procedure should in principle be able to handle quadratic Diophantine equations with any number of variables. (The same is not true of simultaneous quadratic Diophantine equations, and indeed with a vector x of just a few variables, a system m . x^2 == a of such equations could quite possibly show undecidability.) Ever since antiquity there have been an increasing number of scattered results about Diophantine equations involving higher powers. In 1909

39. Stephen Wolfram: A New Kind Of Science | Online (The same is not true of simultaneous quadratic diophantine equations, The vast majority of work on diophantine equations has been for the case of two http://www.wolframscience.com/nksonline/page-1164b-text

Cookies Required A New Kind of Science See http://www.wolframscience.com/nksonlineFAQs.html for more information or send email to support@wolframscience.com Search site Get the NKSwire newsletter Send a Message

40. Diophantine Equation Despite their simple appearance diophantine equations can be fantasticallydifficult to solve. A notorious example comes from Fermat s last theorem http://www.daviddarling.info/encyclopedia/D/Diophantine_equation.html

Return to The Worlds of David Darling INDEX LATEST NEWS ARCHIVE ... Z entire Web this site Diophantine equation An equation that has integer coefficients and for which integer solutions are required. Such equations are named after Diophantus . The best known examples are those from Pythagoras's theorem, a b c , when a b , and c a n b n c n for n > 2. One of the challenges (the tenth one) that David Hilbert threw down to twentieth-century mathematicians in his famous list was to find a general method for solving equations of this type. In 1970, however, the Russian mathematician Yuri Matiyasevich showed that there is no general algorithm for determining whether a particular Diophantine equation is soluble: the problem is undecidable. ALL MATHEMATICS ENTRIES BACK TO TOP var site="s13space1234"

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