81. 11D: Diophantine Equations (Thus the Diophantine equation x^2+y^2=N can be treated both in 11P and here in 11D (as a Pell equation).) Some Diophantine equations are best thought of as part of 11J http://www.math.niu.edu/~rusin/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.

82. Proof Of Fermat's Last Theorem By Kerry M. Evans A proof by Kerry M. Evans. http://fermat.yolasite.com/

In Defense of Mr. Fermat The concern here, of course, is the contradiction in terms of Fermat’s Last Theorem. Many attempts and supposed proofs have preceded by professional Mathematicians and amateurs as well. Invariably, these proposed proofs have met with opposition by nature of inherent dubious assumptions and illicit conclusions. Thus clarity is a matter of first order. Motivation is equally intended. Hopefully, what remains is a matter of logical immediacy. Definition and Constraints * r n = a n b n where Rewriting Exponents What should be considered as the prominent issue is the exclusion of exponent 2 at definition of variables. This value has exceptional properties in the context of eventual generalization and subsequent proof. The method is described as “Rewriting Exponents.” OBJECT: holds for any n such that n is greater than or equal to 3] implies * may be rewritten (in possibly different r, a, b and n) as v n u n w n where n’ is greater than or equal to 2 such that v, u and w are necessarily positive. CASE 1: n has an odd divisor, q, greater than 1.

83. Hilbert's Tenth Problem. Diophantine Equations. By K.Podnieks Given a Diophantine equation with any number of unknowns and with rational integer coefficients devise a process, which could determine by a finite number of operations whether the equation is solvable in rational integers. http://www.ltn.lv/~podnieks/gt4.html

Hilbert tenth problem, Diophantine equation, Hilbert, tenth problem, Matiyasevich, Robinson, Julia, 10th, problem, Davis, Martin, Diophantine, equation Back to title page Left Adjust your browser window Right 4. Hilbert's Tenth Problem Statement of the problem: 10. Determining the solvability of a Diophantine equation. Given a Diophantine equation with any number of unknowns and with rational integer coefficients: devise a process, which could determine by a finite number of operations whether the equation is solvable in rational integers. (See the original statement in German at http://logic.pdmi.ras.ru/Hilbert10/stat/stat.html 4.1. History of the Problem. Story of the Solution Linear Diophantine equations Problems that can be solved by finding solutions of algebraic equations in the domain of integer numbers are known since the very beginning of mathematics. Some of these equations do not have solutions at all. For example, the equation 2x-2y=1 cannot have solutions in the domain of integer numbers since its left-hand side is always an even number. Some other equations have a finite set of solutions. For example, the equation 3x=6 has only one solution x=2. And finally, some equations have an infinite set of integer solutions. For example, let us solve the equation 7x-17y=1:

84. Approaches | Mathematics | Domains | Victorian Essential Learning Standards Sep 16, 2009 A Diophantine equation is one in which only integer coefficients for the terms of the equation and solutions for the variables involved are http://vels.vcaa.vic.edu.au/maths/approaches.html

@import url(/css/vcaa_stylesie.css); @import url(/css/vcaa_tooltip.css); @import url(/css/vcaa_voice.css) aural; Home Contact Us FAQ Glossary ... School Implementation Search Mathematics Domain Introduction Domain Structure Stages of Learning ... Downloads Support Materials Standards and Learning Focus Statements Assessment Maps Progression Point Examples Sample Units DEECD Resources Mathematics Domain Mathematics Developmental Continuum Home Domains ... Mathematics Approaches to Mathematics Introduction Show All Introduction This material provides teachers with several sample learning activities related to the standards for the Structure and Working mathematically dimensions of the Mathematics VELS in particular, and other dimensions as applicable. While Structure does not begin as a dimension with explicit standards until Level 3, the precursor aspects of mathematical structure, that is what students should know and be able to do, in relation to set logic function and algebra are naturally embedded in the other dimensions at Levels 1 and 2. As noted in the introduction to the Mathematics standards: Mathematical reasoning and thinking underpins all aspects of school mathematics, including problem posing, problem solving, investigation and modelling. It encompasses the development of algorithms for computation, formulation of problems, making and testing conjectures, and the development of abstractions for further investigation.

85. 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://algo.inria.fr/csolve/fermat.pdf

86. How To Solve A Linear Diophantine Equation - WikiHow wikiHow article about How to Solve a Linear Diophantine Equation. http://www.wikihow.com/Solve-a-Linear-Diophantine-Equation

87. Equations Involving Sums Of Powers Jerzy Urbanowikz paper on diophantine equations involving sums of powers with quadratic characters as coefficients. http://archive.numdam.org/ARCHIVE/CM/CM_1996__102_2/CM_1996__102_2_125_0/CM_1996

88. The Difficulty Of Schema Conformance Problems File Format PDF/Adobe Acrobat Quick View http://www.unidex.com/scp/paper.pdf

89. Diophantine Equation - Definition Of Diophantine Equation By The Free Online Dic The equation of Fermat''s last theorem is one example of a type known as a Diophantine equation an algebraic expression of several variables whose solutions are required to be http://www.thefreedictionary.com/Diophantine equation

90. 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.

91. HBNweb.de Pythagorean Triples B=(a�/m - M)/2 C=b+m Pythagorean Triplets A JavaScript applet which reads a and gives integer solutions of a^2+b^2 = c^2. http://www.hbnweb.de/pythagoras/pythagoras.html

92. 8 VI. Pell's Equation Provides information on this equation, solved by Brahmagupta in 628 AD. http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch8_6.h

Indian Mathematics MacTutor Index Previous page (8 V. Bhaskaracharya II) Contents Next page (8 VII. The end of the Classic period) 8 VI. Pell's equation Although there were great advancements in many areas of algebra, and the solution of many different forms of equations, from simple forms such as ax c by , to forms as complex as ax bxy cy z , I have chosen to look in slightly more detail at solution(s) to the so called Pell's equation, Nx y . The equation was so called due to a mistake on the part of Euler , who attributed the solution of the equation to John Pell , a 17 th century English scholar, who actually only referred to the equation in a text he wrote on algebra. The equation was nearly solved by Brahmagupta (628 AD) and the solution was improved by Bhaskara II (1150 AD), leading some historians, including C Srinivasiengar, to suggest: ...It is therefore fitting that this equation be called the Brahmagupta-Bhaskara equation. [CS, P 110] The complete theory underlying the solution was expounded by Lagrange in 1767, and rests on the theory of continued fractions. It must be briefly noted how remarkable the achievements of Indian scholars were, given the time period in which equations of the Pell's type were studied. The Indian method involves an element of trail-process but contains no mention of continued fractions. Further to solving equations of the Pell's type to obtain solutions for the unknowns, Brahmagupta extended his method of solution to find square roots. This contribution is of huge interest as it is essentially the same method rediscovered and used by

93. Equal Sums Of Like Powers Dissertation about equal sums of like powers by Tom Womack. http://tom.womack.net/maths/dissert_abstract.htm

Equal Sums of Like Powers - my dissertation This dissertation (written for the MMath degree at Oxford) looks at various problems of the form 'in how many ways can an integer be written as a sum of a given number of positive n th powers'. About the only proven result in this field is that there exist integers writable in arbitrarily many ways as a sum of two positive cubes; I give this result with a proof due to Hardy, and also the results of a large number of searches. In summary: The smallest number writable in five ways as a sum of two positive cubes is 48988659276962496 = 231518 . This number has I believe now been discovered three times independently, by me in December 1998, by D J Bernstein in 1996, and by David W Wilson in November 1997. After I submitted the dissertation, John Cremona discovered using Ian Connell's apecs package that the associated elliptic curve x +y =48988659276962496 in fact has rank 7, which I believe is as high as is known for curves of that form. is the smallest number writable in two ways as a sum of two fourth powers (this was known to Euler); my search gives that, if a number writable in three ways exists, it exceeds 2

94. Google 사전 Translate this page diophantine equation - Google . http://www.google.co.kr/dictionary?hl=ko&sl=en&tl=ko&q=diophantine e

95. Diophantine Set - Wikipedia, The Free Encyclopedia It is not hard to see that every Diophantine set is recursively enumerable consider a Diophantine equation f (n, x 1, , x k) = 0. Now we make an algorithm which simply tries all http://en.wikipedia.org/wiki/Diophantine_set

Diophantine set From Wikipedia, the free encyclopedia Jump to: navigation search In mathematics , a Diophantine set of j tuples of integers is a set S for which there is some polynomial with integer coefficients f n n j x x k such that a tuple n n j of integers is in S if and only if there exist some (non-negative) integers x x k with f n n j x x k Such a polynomial equation over the integers is called a Diophantine equation . In other words, a Diophantine set is a set of the form where f is a polynomial function with integer coefficients. Matiyasevich's theorem , published in 1970, states that a set of integers is Diophantine if and only if it is recursively enumerable . A set S is recursively enumerable precisely if there is an algorithm that, when given an integer, eventually halts if that input is a member of S and otherwise runs forever. This means that the concept of general Diophantine set, apparently belonging to number theory , can be taken rather in logical or recursion-theoretic terms. This is far from obvious, however, and represented the culmination of some decades of work. Matiyasevich's theorem effectively settled Hilbert's tenth problem . It 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 has a solution among the integers.

96. Power Page Steve Dutch s page about powers of numbers. http://www.uwgb.edu/dutchs/RECMATH/rmpowers.htm

Power Page Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay First-time Visitors: Please visit . Use "Back" to return here. Interesting stuff about powers of numbers Index About Notation Pythagorean Triplets Almost-Isoceles Triples Almost 30-60 Triples ... Multigrades About Notation To save writing exponents, we'll denote power sums like this: terms on each side of the equality will be separated by commas, and opposite sides of the equality will be separated by a semicolon. Thus, for Pythagorean triplets, a, b; c means a + b = c . For chains of squares, 21, 22, 23, 24; 25, 26, 27 means 21 and similarly for sums of higher powers. Where the exponent is not clear from context, it will be noted in parentheses. Thus: 95,800, 217,519, 414,560; 422,481 (n=4) means 95,800 Pythagorean Triplets The simplest triangle with integer sides that satisfies the Pythagorean Theorem is 3, 4, 5. That is, 3 . Such a triplet is called a Pythagorean Triplet . There are infinitely many, and they are easy to generate. A classic formula, known since ancient times, can generate them at will. If the numbers in the triplet are a, b, and c, then: a = n -m , b=2mn, c=m +n , where m and n are two integers and m is less than n.

97. CiteSeerX � � RELAX NG With W3C XML Schema Datatypes by RC Lyons 2003 http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.102.414

98. Sortedsums D. J. Bernstein s collection of tools for enumerating small solutions to certain types of equal sums of like powers. http://cr.yp.to/sortedsums.html

D. J. Bernstein More number-theoretic computations sortedsums The latest published sortedsums package is sortedsums-0.53.tar.gz I have a paper explaining the algorithms used here. What is it? solvepqrs enumerates (i,j,k,l) such that p[i]+q[j] = r[k]+s[l], in increasing order of p[i]+q[j]. solvepppp, solvepprs, solvepqpq, and solvepprr are faster versions of solvepqrs. Several sample applications are included in the sortedsums package: cubepqrs enumerates solutions to pa^3 + qb^3 = rc^3 + sd^3, given (p,q,r,s). cubepppp, cubepprs, and cubepqpq are faster versions of cubepqrs. two3 enumerates positive integers that can be written in many ways as sums of two cubes. two3p enumerates integers that can be written in many ways as sums of two positive cubes. two4 enumerates integers that can be written in two ways as sums of two positive fourth powers. euler4 enumerates fourth powers that can be written as sums of three positive fourth powers. This work was supported by the National Science Foundation under grant DMS-9600083. Results I have a table The fourth power of 8707481 is a sum of three positive fourth powers. Same for 12197457, 16003017, and 16430513. Other small known solutions are 422481 (Frye), 2813001 (MacLeod), 20615673 (Elkies), and 638523249 (MacLeod). There are no other primitive solutions below 21000000. This computation used

99. Diophantine Equation - Definition And More From The Free Merriam-Webster Diction Definition of word from the MerriamWebster Online Dictionary with audio pronunciations, thesaurus, Word of the Day, and word games. http://mw1.merriam-webster.com/dictionary/diophantine equation

100. Mathematician's Secret Room Unsolved problems in Number Theory. English and Japanese text by Hisanori Mishima. http://www.asahi-net.or.jp/~KC2H-MSM/mathland/overview.htm

Mathematician's Secret Room Challenges to Unsolved Problems in Number Theory Contents What's new (July 22, 2009) Chapter 0 : Opening - Why I am interested in Number Theory. Chapter 1 : 4/n = 1/a + 1/b + 1/c [D11] Chapter 2 : Squares consisted of 3 different digits [F24] Chapter 3 : n = (x + y + z)(1/x + 1/y + 1/z) Chapter 4 : n = x + y + z Chapter 5 : Repeating Decimals [A3] Chapter 6 : Additive Palindromicness of Natural Numbers [F32] Chapter 7 : Collatz's Conjecture [E16] Chapter 8 : Continued Fraction and Pell's Equation Chapter 9 : Amicable Numbers [B4][B5] Chapter 10 : Congruent Numbers (Congruum) [D27] Chapter 11 : Number Theoretic Algorithms [A3] [B45] [F10] Chapter 12 : Integer Factorization Algorithms Appendix 1 : WIFC (World Integer Factorization Center) Appendix 2 : Bibliography Appendix 3 : How to join in the Factorization Project Appendix 4 : Benchmark of Tomabechi's "ppmpqs.exe" Appendix 5 (new !) Prime table (Primes near to 2 n n , factorial, primorial and compositorial) [nn] denote the number in Richard K. Guy

Page 5     81-100 of 145    Back | 1  | 2  | 3  | 4  | 5  | 6  | 7  | 8  | Next 20