61. LinuxGuruz Foldoc Page Want to make a donation? $. LinuxGuruz Foldoc. diophantine equation. mathematics Equations with integer coefficients to which integer solutions are sought. http://foldoc.linuxguruz.org/foldoc.php?Diophantine equation

62. Equal Sums Of Like Powers counterexample 144 5 = 27 5 + 84 5 + 110 5 + 133 5. diophantine equation Ageneral method exists for first degree diophantine equation. However http://member.netease.com/~chin/eslp/collect.htm

Equal Sums of Like Powers Internet Resources Collection on Equal Sums of Like Powers The Prouhet, Tarry, Escott problem revisited Peter Borwein pborwein@cecm.sfu.ca ) and Colin Ingalls Enseign.Math. (2).40.(1994),no.1-2,3-27, MR95d:11038 Abstract. The author gives a collection of known elementary result, and discusses the most interesting case n=k+1.Some open problems are present. Computing Minimal Equal Sums Of Like Powers Jean-Charles Meyrignac's distributed-computing project on equal sums of like powers and the place to look for the current status of the problem. New results in equal sums of like powers Randy L. Ekl's article in Math. Comp. 67 (1998), pp. 1309-1315. The Tarry-Escott multigrades problem Abstract. Given a positive integer n, find two sets of integers a_1, ..., a_r and b_1, ..., b_r, with r as small as possible, such that sum (a_j)^k = sum (b_j)^k for k = 1, 2, ..., n. Conjecture: r=n+1 for all n. Pure product polynomials and the Prouhet-Tarry-Escott problem Roy Maltby Math. Comp. (1997), pp. 1323-1340 Abstract.

63. Nikolaos G. Tzanakis List of Publications 1. The diophantine equation x 3 + 3y 3 = 2 n , J.Number Th.15 (1982), 376387. 6. On the diophantine equation 2x 3 + 1 = py 2 , Manuscr. http://www.uch.gr/Tmhmata/MATHEMATICS_OLD/Tzanakis.html

Nikolaos (Nikos) Tzanakis Date and place of birth: July 7,1952, Iraklion, Crete, Greece. Academic title: Professor. Area of research: Number Theory (Explicit solution of diophantin equation ). Ph.D.: Department of Mathematics, University of Athens, Greece, 1981. List of Publications: 1. The diophantine equation x n J.Number Th. 15 2. On the diophantine equation y - D = 2 k J.Number Th. 3. (In cooperation with A.Bremner) Integer points on y = x Math.Comp. 4. The diophantine equation x - y = 1 and related equations, J .Number Th. 5. The complete solution in integers of x n J. Number Th. 6. On the diophantine equation 2x + 1 = py Manuscr. Math. 7. A remark on a theorem of W.E.H.Berwick, Math. Comp. 8. On the diophantine equation x - Dy = k , Acta Arithm. 9. (In cooperation with J.Wolfskill) On the diophantine equation y n J. Number Th. 10. (In cooperation with J.Wolfskill) The diophantine equation y a/2 + 4q + 1 with an application to Coding Theory

64. 11D: Diophantine Equations Dave Rusin's guide to diophantine equations.Category Science Math Number Theory diophantine equations...... of squares and so on. (Thus the diophantine equation x^2+y^2=N canbe treated both in 11P and here in 11D (as a Pell equation).). http://www.math.niu.edu/~rusin/papers/known-math/index/11DXX.html

Search Subject Index MathMap Tour ... Help! ABOUT: Introduction History Related areas Subfields POINTERS: Texts Software Web links Selected topics here 11D: Diophantine equations Introduction History Applications and related fields See also 11GXX, 14GXX. In particular, discussion of many examples and families of equations has been moved to pages for (arithmetic) algebraic geometry; the dividing line is unclear sorry. Diophantine equations whose solution set is one-dimensional are discussed with algebraic curves . This includes single equations in 2 variables (or homogeneous equations in 3 variables, such as the Fermat equation). In particular,... Equations whose solutions are curves of genus 1 are discussed in the subsection on elliptic curves . Examples include cubics in two variables, homogeneous cubics in three variables, pairs of quadratics in four variables, and equations of the form y^2=Q(x) where Q is a polynomial of degree 3 or 4. Sets of N equations in N+2 variables (or N+3 variables, if those equations are homogeneous) describe algebraic surfaces ; for example the question of the existence of a "rational box" is there.

65. Gödel's Theorem A diophantine equation is an equation of the form p(x 1 ,..x n )=0,where p(x 1 ,..x n ) is a polynomial with integer coefficients. http://www.sm.luth.se/~torkel/eget/godel/self.html

Is self-reference essential to Gödel's theorem? The original proof of Gödel's theorem uses the so-called Gödel sentence G for a theory T. This sentence is usually described as a sentence which says of itself that it is unprovable in T, or as a formalization of "I am unprovable in T" or "this sentence is unprovable in T". The Gödel sentence is in fact self-referential in the specific sense that it has the form A(t), where it is provable in a weak arithmetical theory that the value of the term t is the Gödel number of the formula A(t) itself. As a consequence, a certain equivalence is a theorem of T. This equivalence has the form "G holds if and only if the formula satisfying ... is not a theorem of T", where the formula satisfying ... is in fact the formula G itself. This equivalence is what is used in Gödel's proof of the incompleteness theorem. Thus it makes good sense to say that the Gödel sentence G is self-referential, and there is no mystery about how sentences that are in this sense self-referential can be constructed (in various ways) for different theories T. Gödel's original proof of his theorem, using the Gödel sentence G, is vivid and memorable, but it has given many people the incorrect impression that G must be used to prove the incompleteness theorem for T, or that sentences undecidable in T must be in this sense self-referential.

66. Number Theory: Diophantine Equations A diophantine equation is an algebraic equation in one or more unknownswith integer coefficients, for which integer solutions are sought. http://uzweb.uz.ac.zw/science/maths/zimaths/62/dioph.htm

Number Theory: Diophantine Equations Introduction Integers have gradually lost association with superstition and mysticism, but their interest for mathematicians has never waned. Among the greatest mathematicians is Diophantus of Alexandria (275 A D), an early algebraist, sometimes called `The Father of Algebra'. He left his mark on the theory of numbers, and his name - there are Diophantine numbers, and Diophantine equations. Diophantine Equations The Euclidean algorithm for finding the greatest common divisor of two integers leads to an important method for representing the quotient of two integers as a composite fraction. For example, applied to 840 and 611, the Euclidean algorithm yields the series of equations: which incidently shows that the greatest common divisor (840, 611) = 1. From these equations, we have derived the following expressions: Combining these operations, we obtain the development of the rational number [ 840/611] in the form An expression of the form a = a a a :+ [ 1/(a n + [ 1/(a n where the a's are positive integers, is called a

67. Biography Of Diophantus Diophantus' Work. diophantine equations. A diophantine equation is a polynomial equationwith integral coefficients to which only integral solutions are sought. http://www.andrews.edu/~calkins/math/biograph/biodioph.htm

Back to Table of Contents Biographies of Mathematicians - Diophantus Diophantus's Life Born: about 200 A.D. in Alexandria , Egypt Died: about 284 A.D. in Alexandria , Egypt Diophantus worked during the middle of the third century and is best known for his Arithmetica , a work on the theory of numbers. Little is known of Diophantus's life. The most details we have (and these may not be accurate) say that he married at the age of 33 and had a son who died at the age of 42, four years before Diophantus himself died at approximately 84. Diophantus's epitaph The most details we have found of Diophantus's life have come from Greek Anthology epigrams. Which is a collection of number games and strategy puzzles. Because of the speculation of his age the only significant evidence of his lifespan is through his collection of puzzles. "This tomb hold Diophantus. Ah, what a marvel! And the tomb tells scientifically the measure of his life. God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after a seventh, and in the fifth year after his marriage He granted him a son. Alas! late-begotten and miserable child, when he had reached the measure of half his father's life, the chill grave took him. After consoling his grief by this science of numbers for four years, he reached the end of his life." J R Newman (ed.) The World of Mathematics (New York 1956).

68. Biography Of Diophantus Diophantus' Work. A diophantine equation is a polynomial equation withintegral coefficients to which only integral solutions are sought. http://www.andrews.edu/~calkins/math/biograph/199899/biodioph.htm

Back to Table of Contents Biographies of Mathematicians - Diophantus Diophantus's Life Born: about 200 A.D. in Alexandria , Egypt Died: about 284 A.D. in Alexandria , Egypt Diophantus worked during the middle of the third century and is best known for his Arithmetica , a work on the theory of numbers. Little is known of Diophantus' life. The most details we have (and these may not be accurate) say that he married at the age of 33 and had a son who died at the age of 42, four years before Diophantus himself died at approximately 84. Diophantus's epitaph "This tomb hold Diophantus. Ah, what a marvel! And the tomb tells scientifically the measure of his life. God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after a seventh, and in the fifth year after his marriage He granted him a son. Alas! late-begotten and miserable child, when he had reached the measure of half his father's life, the chill grave took him. After consoling his grief by this science of numbers for four years, he reached the end of his life." J R Newman (ed.) The World of Mathematics (New York 1956).

69. CTK Exchange The Knot, Recommend this page. Subject A diophantine equation DateFri, 12 Sep 1997 183403 +0800 From Calvin Lui. Hi ,I am Ronald http://www.cut-the-knot.com/exchange/diophantine.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Subject: A diophantine equation Date: Fri, 12 Sep 1997 18:34:03 +0800 From: Calvin Lui Hi ,I am Ronald.I have a mathematics question.Could you help me to solve it? Question: For A do not equal to zero , A B C D E is five different integers Find: all the value of A ,B ,C, D, E which statisfy the following equation: A^(BCDE)= A(B^2)(C^3)(D^4)(E^5) Thank you! Alexander Bogomolny G o o g l e Web Search Latest on CTK Exchange embedding applets Posted by Stuart Carduner 1 messages 09:05 AM, Feb-19-03 math problem no answer in sight Posted by Dmom 4 messages 10:40 AM, Feb-28-03 anticentre+argand diagrams Posted by Taka 0 messages 07:02 PM, Mar-15-03 Tiling hyperspheres Posted by Graham C 4 messages 10:38 AM, Mar-12-03 archive says feb instead of march Posted by Frank Anzalone 1 messages 01:15 AM, Mar-06-03 Probability Posted by Ralph scalitzia 1 messages 07:03 PM, Mar-15-03

70. Publications Translate this page Compositio Math. 132 (2002), 137-158. (avec TN Shorey) On the diophantine equation(x^m - 1)/(x - 1) = (y^n-1)/(y-1), Pacific J. Math. 207 (2002), 61-75. http://www-irma.u-strasbg.fr/~bugeaud/publi.html

Publications Sets of exact approximation order by rational numbers. Mahler's classification of numbers compared with Koksma's. Nombres de Liouville et nombres normaux. C. R. Acad. Sci. Paris, Ser. I 335 (2002), 117-120. On U_m numbers with small transendence measure. A note on inhomogeneous Diophantine approximation. (avec P. Corvaja et U. Zannier) An upper bound for the G.C.D. of a^n - 1 and b^n - 1. Math. Z. 243 (2003), 79-84. J. Number Theory 96 (2002), 174-200. (avec A. Dujella) On a problem of Diophantus for higher powers. Approximation by algebraic integers and Hausdorff dimension. J. London Math. Soc. 65 (2002), 547-559. Linear forms in two m -adic logarithms and applications to Diophantine problems. Compositio Math. 132 (2002), 137-158. (avec T. N. Shorey) On the Diophantine equation (x^m - 1)/(x - 1) = (y^n-1)/(y-1), Pacific J. Math. 207 (2002), 61-75. (avec G. Hanrot et M. Mignotte) Proc. London Math. Soc. 84 (2002), 59-78. Archiv Math. 79 (2002), 34-38. (avec T. N. Shorey) On the number of solutions of the generalized Ramanujan-Nagell equation.

71. Diophantine Equations Information Sites 3, 8, 120. Developing A General 2nd Degree diophantine equation x^2+ p = 2^n Methods to solve these equations. Thue Equations http://numbersorg.com/NumberTheory/DiophantineEquations/

NUMBERSorg.com Search SPYorg.com (Not sure of spelling? Use first letters and * such as abc* or abcd* or abcde*) Match:.. All Any Format: Long Short Search Words: Top Science Math Number Theory : Diophantine Equations Equal Sums of Like Powers Fermat's Last Theorem Linear Diophantine Equations - A web tool for solving Diophantine equations of the form ax + by = c. Bibliography on Hilbert's Tenth Problem - Searchable, ~400 items. Diophantine Equations - Dave Rusin's guide to Diophantine equations. Egyptian Fractions - Lots of information about Egyptian fractions collected by David Eppstein. The Erdos-Strauss Conjecture - 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. Developing A General 2nd Degree Diophantine Equation x^2 + p = 2^n - Methods to solve these equations. Thue Equations - Definition of the problem and a list of special cases that have been solved, by Clemens Heuberger. Hilbert's Tenth Problem - Statement of the problem in several languages, history of the problem, bibliography and links to related WWW sites. Diophantine Geometry in Characteristic p - A survey by Jos© Felipe Voloch.

72. A Message Transfer Method Using Powers Mod A Prime Theorem If p is a prime and then . A diophantine equation is an equationwhere one is only interested in solutions which are integers. http://www.mathlab.cornell.edu/computer_and_portfolio/discrete/prime_power/

Next: Some Things to Try A Message Transfer Method Using Powers mod a Prime Math Explorers Club October 14, 2000 This computer activity involves securely transferring a message x from a Alice to Bob using a publically know prime number p . The channel between Alice and Bob is assumed to possibly have eavesdroppers, and the idea is to make it hard for the eavesdroppers to decode the message. The message is assumed to have already been converted to an integer between 1 and p Here's the algorithm (sometimes known as the Massey-Omura cryptosystem): Pick Some Primes: Both Alice and Bob privately pick some large primes e A and e B respectively. Each also checks that their primes have no common factor with p -1. (Here p is the publically known prime.) Solve Some Diophantine Equations: Alice privately finds a number d A so that . Similarly Bob finds d B so that Alice's Step 1: Compute and transmit m eA to Bob. Here the subscript eA is a way of denoting which operation ( e) and who ( A for Alice) has applied the operation. Bob's Step 1: Compute and transmit m eAeB to Alice.

73. Bibi Drum DETERMINATION OF THE SOVABILITY OF A diophantine equation. A diophantine equationis an polynomial equation in which only integer solutions are allowed. http://www.cs.appstate.edu/~sjg/womenandminoritiesinmath/student/robinson/robins

Bibi Drum Joy Winstead March 29, 2001 Julia Bowman Robinson is no longer with us, though her work survives through the efforts of many dedicated people. Julia was born on December 8, 1919 in St. Louis Missouri. As a child she faced a series of traumatic events, which affected her throughout her life. At the age of two, Bowman’s mother passed away. Julia and her sister, Constance, were sent with their nurse to Arizona to live with their grandmother. Their father owned a machine tool and equipment company, and soon after the death of his wife, lost interest in the business. Along the way he had saved up substantial amount of money and felt that he had enough income to support a family. Julia’s father remarried and closed down his business in order to move to Arizona and live with Julia and Constance. At the age of nine, Julia came down with scarlet fever. It was because of this, that she was isolated from her family. During this time her father took care of her. Shortly after Julia recovered from scarlet fever, she came down with rheumatic fever and spent an entire year in bed. Because of these illnesses, Julia missed more than two years of school.

74. CRC Concise Encyclopedia Of Mathematics On CD-ROM: D Dini's Test; Dinitz Problem; Diocles's Cissoid; diophantine equation;diophantine equation5th Powers; diophantine equation6th Powers; http://www.math.pku.edu.cn/stu/eresource/wsxy/sxrjjc/wk/Encyclopedia/math/d/d.ht

d'Alembert's Equation d'Alembert Ratio Test d'Alembert's Solution d'Alembert's Theorem ... Dynkin Diagram Eric W. Weisstein

75. Solving Linear Diophantine Equations Linear diophantine equations Let a,b and c be integers (a 0 b). Theexpression ax + by = c is called a Linear diophantine equation. http://home.cc.umanitoba.ca/~umbrow45/linear_diophantine.html

Solving Linear Diophantine Equations (By Hand) This document might be of interest if you're taking 74.213, Discrete Mathematics for Computer Science. We had to know this for both exams; no calculators, notes or any aids were allowed. Some of this material was gathered from my class notes (regular session 2000), some from other sources. Fonts used in this document are arial. Solving these equations is not in the textbook, so I decided to post it. Linear Diophantine equations: Let a,b and c be integers (a b). The expression ax + by = c d = gcd(a,b). we have an integer k such that k = c/d A particular solution of the equation is (x p ,y p where x p = kx`, and y p = ky`, with x` = (-1) n-1 B n-1 , and y` = (-1) n A n-1 A general solution of the equation is x = x p - (b/d)t y = y p + (a/d)t t Z (Gives all solutions. Remember all these equations in bold and how to do the table!) Example. Solve 121x - 88y = 572: Solve for (x, -y) [negate y solution at the end] We use A n-1 and B n-1 computed using the q’s (quotients) from the Euclidean Algorithm: A A A n+1 = q n+1 A n + A n-1 B B B n+1 = q n+1 B n + B n-1 Compute gcd(a,b) using the Euclidean Algorithm:

76. NRICH Mathematics Enrichment Club (1293.html) The Nrich Maths Project Cambridge, England. Mathematics resources for children, parents and teachers to enrich learning. Published on the 1st of each month. Problems, children's solutions, interactivities, games, articles, news http://nrich.maths.org/askedNRICH/edited/1293.html

Asked NRICH NRICH Prime NRICH Club ... Get Printable Page March 03 Magazine Site Update News Events Problems Solutions ... Games Archive Problems Solutions Articles Inspirations ... Interactivities Web board Ask NRICH Asked NRICH NRICH Club Register Tough Nuts About Help! ... Where is NRICH? Associated Projects Maths Thesaurus MOTIVATE EuroMaths Millennium Maths ... Project Display maths using fonts images Help Back Issues Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Bernard's Bag(P) - solutions(P) Penta Probs(P) - solutions(P) Let Me Try(P) - solutions(P) Kid's Mag(P) Play Games(P) Staff Room(P) 6 Problems - solutions 15+Challenges - solutions Articles Games LOGOland Editorial News Diophantine equation By Carol Toogood (t314) on April 20, 1998 Prove there are no integer solutions to x Thanks By Gareth McCaughan on April 24, 1998 The possible values for x mod 13 are 0,1,5,8,12. The possible values for y mod 13 are 0,1,3,9, so the possible values for 2y are 0,2,6,5, so the possible values for 2y that are possible for the LHS and possible for the RHS, so there are no solutions to the equation mod 13.

77. Unsolved Problem 18 30Apr-1995 Unsolved Problem 18 Are there distinct positive integers,a, b, c, and, d such that a^5+b^5=c^5+d^5? It is known that http://cage.rug.ac.be/~hvernaev/problems/Proble18.html

30-Apr-1995 Unsolved Problem 18: Are there distinct positive integers, a, b, c, and, d such that a^5+b^5=c^5+d^5? It is known that 1^3+12^3=9^3+10^3 and 133^4+134^4=59^4+158^4, but no similar relation is known for fifth powers. Other remarkable identities are 27^5+84^5+110^5+133^5=144^5 and 2682440^4+15365639^4+18796760^4=20615673^4. Reference: [Guy 1994] Richard K. Guy, Unsolved Problems in Number Theory, second edition. Springer Verlag. New York: 1994. Page 140. Each week, for your edification, we publish a well-known unsolved mathematics problem. These postings are intended to inform you of some of the difficult, yet interesting, problems that mathematicians are investigating. We do not suggest that you tackle these problems, since mathematicians have been unsuccessfully working on these problems for many years. general references previous problem problem archive next problem ... electronic publications

78. The Prime Glossary: Logarithmic Function logarithmic function (another Prime Pages' Glossary entries). The Prime Glossary. http://primes.utm.edu/glossary/page.php?sort=Log

79. Home Page Of Nikos Tzanakis ?p?st?µ µ?s?e?se 1 The diophantineequation x^3 + 3y^3 = 2^n , J.Number Th. 15 (1982), 376387. http://itia.math.uch.gr/~tzanakis/indexg.html

English Page ñáöåßï: H 303 tzanakis@math.uch.gr The diophantine equation x^ n J.Number Th On the diophantine equation y^ - D = 2^ k J.Number Th ] (In cooperation with A.Bremner) Integer points on y^ = x^ Math.Comp The diophantine equation x^ - y^ = 1 and related equations J .Number Th The complete solution in integers of x^ n J. Number Th On the diophantine equation 2x^3 + 1 = py^2 Manuscr. Math A remark on a theorem of W.E.H.Berwick Math. Comp On the diophantine equation x^2 - Dy^4 = k Acta Arithm ] (Óõíåñãáóßá ìå ôïí J.Wolfskill) On the diophantine equation y^2 = 4q^n + 4q + 1 J. Number Th ] (Óõíåñãáóßá ìå ôïí J.Wolfskill) The diophantine equation y^ a/2 + 4q + 1 with an application to Coding Theory J. Number Th ] (Óõíåñãáóßá ìå ôïí R. J.Stroeker) On the application of Skolem's p-adic method to the solution of Thue equations J.Number Th ] (Óõíåñãáóßá ìå ôïí B.M.M. de Weger) On the practical solution of the Thue equation J.Number Th On the practical solution of the Thue equation - An outline Colloq. Math. Soc. Janos Bolyai

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