21. Mathsoft: Mathsoft Unsolved Problems: 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

search site map about us  + news  + ... Unsolved Problems Links On a Generalized Fermat-Wiles Equation Zero Divisor Structure in Real Algebras Sleeping Habits of Armadillos Mathsoft Constants Engineering Standards ... Math Resources On a Generalized Fermat-Wiles Equation Fermat's Last Theorem was no more than a conjecture for over 350 years. Let n be an integer greater than 2. Fermat claimed that any integers x, y and z, not necessarily positive, for which x n + y n = z n To some people, the passage of this conjecture to theoremhood is marked by sadness. They may mistakenly believe that no other interesting Diophantine equations are left to be solved. This essay is aimed at such individuals: there is a much larger class of equations, of which Fermat-Wiles is only a special case, that is well worth everyone's attention! The equation we'll examine is x n + y n n where c is a positive integer. We wish to learn what conditions on n and c force the existence of a non-trivial 0. In other words, when is the equation x n + y n n solvable x + y Click here for more terms The infinite sequence of all such integers c can be shown to be highly sparse relative to the sequence of positive integers. By the

22. LinearDiophantineEquation Linear diophantine equation (English). Search for Linear Diophantineequation in NRICH PLUS maths.org Google. Definition level 2. http://thesaurus.maths.org/mmkb/entry.html?action=entryById&id=888

23. 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.

24. Diophantine Equation diophantine equation. Introduction. A diophantine equation is an equation in which only Integer solutions are allowed. Hilbert's 10th Problem asked if a technique for solving a general Diophantine existed. http://granite.sru.edu/~venkatra/Diophantine_Equation.htm

25. 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

26. Diophantine Equation diophantine equation 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 http://rdre1.inktomi.com/click?u=http://planetmath.org/encyclopedia/DiophantineE

27. Diploma Thesis Stoll Thomas) (ps) The diophantine equation X^MY^N=1, known as Catalan s equation,can well be treated with Baker s theory of linear forms in logarithms. http://finanz.math.tugraz.at/~stoll/diplthesis.htm

Diploma Thesis Algorithmic Solution of Diophantine Equations (Stoll Thomas) ( ps The Diophantine equation X^M-Y^N=1, known as Catalan's equation, can well be treated with Baker's theory of linear forms in logarithms. In detail, we give both Baker's proof of his general transcendence result and Tijdeman's proof on Catalan's equation, which gives an explicit upper bound for max(x,y,m,n). This method for showing effective finiteness also works for elliptic and hyperelliptic equations. In the thesis much emphasize is laid on the decidability of non-effective finiteness of the more general Diophantine equation F(X)=G(Y), F and G being arbitrary polynomials. Recent progress has been made by Bilu and Tichy in showing that, by making explicit Siegel's theorem, these polynomials have to satisfy polynomial identities with the o-called standard pairs. In the sequel, an algorithmic implementation of such criterion is given as a Maple worksheet. We study the case, where F and G belong to a polynomial family. For this reason we include a general approach to define orthogonal polynomials and state the fundamental properties they all have in common. In the case where F and G are Meixner polynomials of fixed degree, we prove that there are only finitely many integral solutions (X,Y) to F(X)=G(Y). Geometrically speaking, two octahedrons of dimensions N and M, respectively, can have equally many integral points just in a finite number of cases. thesis advisor: Tichy Robert, Dr.phil., O.Univ.-Prof.

28. Historical Notes: Diophantine Equations Particularly from the work of Carl Friedrich Gauss around 1800 there emergeda procedure to find solutions to any quadratic diophantine equation in two 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

29. Stephen Wolfram: A New Kind Of Science | Online diophantine equations. Any algebraic b. Even the simplest quadratic Diophantineequations can already show much more complex behavior. The http://www.wolframscience.com/nksonline/page-944f-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 Sign the Guestbook

30. CWI Tract CWI Tract 65. Algorithms for diophantine equations. by BMM de Weger. Then it becomesfeasible to determine all the solutions of the original diophantine equation. http://www.cwi.nl/publications/Abstracts_tracts/tr-65.html

CWI Tract CWI Tract 65 Algorithms for diophantine equations by B.M.M. de Weger order form order via WWW Comments to

31. Math Forum - Ask Dr. Math Archives: Diophantine Equations Linear diophantine equation Given positive integers a and b such that a b^2,b^2 a^3, a^3 b^4, b^4 a^5, . prove that a = b. Solving Multivariable http://mathforum.org/library/drmath/sets/select/dm_diophantine.html

Ask Dr. Math Diophantine Equations Dr. Math Home Elementary Middle School High School ... Dr. Math FAQ Diophantine Equations , a selection of answers from the Dr. Math archives. Diophantine Equations We have searched the Web for information about Diophantine equations. Diophantine Equations, Step by Step Find all positive integer solutions to 43x + 7y + 17z = 400. Integer Solutions of ax + by = c Given the equation 5y - 3x = 1, how can I find solution points where x and y are both integers? Also, how can I show that there will always be integer points (x,y) in ax + by = c if a, b and c are all integers? Buying Cows, Pigs, and Chickens A farmer buys 100 animals for $100. The animals include at least one cow, one pig, and one chicken. If a cow costs $10, a pig costs $3, and a chicken costs $0.50, how many of each did he buy? How Many Mice, Cats, and Dogs? You must spend $100 to buy 100 pets, choosing at least one of each pet. The pets and their prices are: mice @ $0.25 each, cats @ $1.00 each, and dogs @ $15.00 each. How many mice, cats, and dogs must you buy? Money Puzzle A man goes to the bank and asks for x dollars and y cents.

32. Math Forum - Ask Dr. Math A diophantine equation is one equation in at least two variables, say x andy, whose solutions (x,y) are required to be whole numbers (integers). 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/

33. Title Details - Cambridge University Press Home Catalogue Exponential diophantine equations. Related Areas Pure Mathematics.Exponential diophantine equations. TN Shorey, R. Tijdeman. £50.00. http://titles.cambridge.org/catalogue.asp?isbn=0521268265

34. Title Details - Cambridge University Press Home Catalogue diophantine equations over Function Fields. Related Areas PureMathematics. diophantine equations over Function Fields. RC Mason. £16.99. http://titles.cambridge.org/catalogue.asp?isbn=0521269830

35. Diophantine Equation diophantine equation. diophantine equations rational values. Examplesof diophantine equations are ax + by = 1 See B?out s identity. http://www.fact-index.com/d/di/diophantine_equation.html

Main Page See live article Alphabetical index Diophantine equation Diophantine equations are equations of the form f = 0, where f is a polynomial with integer coefficients in one or several variables which take on integral values. They are named after Diophantus who studied equations with variables which take on rational values. Examples of Diophantine equations are ax by = 1: See Bézout's identity x n y n z n : 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. x n y Pell's equation ) which is named, mistakenly, after the English mathematician John Pell. It was studied by Fermat The depth of the study of general Diophantine equations is shown by the characterisation of Diophantine sets as recursively enumerable The field of Diophantine approximation deals with the cases of Diophantine inequalities : variables are still supposed to be integral, but some coefficients may be irrational numbers, and the equality sign is replaced by upper and lower bounds. This article is from Wikipedia . All text is available under the terms of the GNU Free Documentation License

36. Diophantine Equations -- From Eric Weisstein's Encyclopedia Of Scientific Books diophantine equations. see also diophantine equation. Bashmakova, Isabella Grigoryevna.Diophantus and diophantine equations. Washington, DC Math. http://www.ericweisstein.com/encyclopedias/books/DiophantineEquations.html

Diophantine Equations see also Diophantine Equation Bashmakova, Isabella Grigoryevna. Diophantus and Diophantine Equations. Washington, DC: Math. Assoc. Amer., 1997. 104 p. $21.95. Beiler, A.H. Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, 1966. A great place to start learning number theory. Well worth the modest price. $7.95. Carmichael, Robert Daniel. The Theory of Numbers, and Diophantine Analysis. New York: Dover, 1959. An unabridged and unaltered republication of the two works, originally published separately. Out of print. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 96-104, 1965. $7.95. Hardy, G.H. and Wright, E.M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979. $42.95. Hunter, J.A.H. and Madachy, J.S. ``Diophantos and All That.'' Ch. 6 in Mathematical Diversions. New York: Dover, pp. 52-64, 1975. $5.95. Ireland, K. and Rosen, M. ``Diophantine Equations.'' Ch. 17 in A Classical Introduction to Modern Number Theory, 2nd ed.

37. PlanetMath: Diophantine Equation diophantine equation, (Definition). A diophantine equation is an equationfor which the solutions are required to be integers. Generally 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 ... Corrections talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information Docs Classification News Legalese ... TODO List 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:

38. Research.html (dvi, pdf). On the diophantine equation G_ {n}(x)=G_{m}(P(x)), joint work with A.Pethõ and RF Tichy, Monatsh. Math. 137 (2002) 3, 173196 Preprint (dvi, pdf). http://finanz.math.tu-graz.ac.at/~fuchs/research.html

Research Interests Number Theory: - Diophantine Equations - Subspace Theorem of W.M. Schmidt - Diophantine m-Tuples Algebraic Geometry: - Algebraic Function Fields and Applications Coding Theory: - Geometric Goppa-Codes - AG-Codes due to Xing, Niederreiter, Lam Publications Algebraisch-geometrische Codes , Diplomarbeit, TU Wien, 2000. ( dvi pdf Quantitative finiteness results for Diophantine equations, PhD-Thesis (Dissertation), TU Graz, 2002. ( dvi pdf joint work with and R. F. Tichy Monatsh. Math. Preprint: ( dvi pdf Diophantine m-tuples for linear polynomials, joint work with A. Dujella and R. F. Tichy Period. Math. Hungar. Preprint: ( dvi pdf Perfect powers in linear recurring sequences, joint work with R. F. Tichy

39. Clemens Heuberger - Thue Equations Among his 23 Problems, Hilbert raised the question, whether there exists an algorithmto solve any given polynomial diophantine equation; the negative answer 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.

40. Diophantine Equation Previous diode Next DIP. diophantine equation. mathematics Equationswith integer coefficients to which integer solutions are sought. http://burks.brighton.ac.uk/burks/foldoc/28/32.htm

The Free Online Dictionary of Computing ( http://foldoc.doc.ic.ac.uk/ dbh@doc.ic.ac.uk Previous: diode Next: DIP Diophantine equation mathematics integer coefficients to which integer solutions are sought. Because the results are restricted to integers, different algorithms must be used from those which find real solutions. [More details?]

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