61. Diophantine Geometry In Characteristic P: A Survey A survey by Jos Felipe Voloch. http://www.ma.utexas.edu/users/voloch/surveylatex/surveylatex.html

Next: Introduction Diophantine geometry in characteristic p: a survey ... it goes without saying that the function-fields over finite fields must be granted a fully simultaneous treatment with number-fields, instead of the segregated status, and at best the separate but equal facilities, which hitherto have been their lot. That, far from losing by such treatment, both races stand to gain by it,... Updates Introduction Curves Abelian varieties and their subvarieties ... About this document ... Felipe Voloch Tue Jan 16 16:22:19 CST 1996

62. Dario Alpern's Generic Two Integer Variable Equation Solver Dario Alpern s Java/JavaScript code that solves Diophantine equations of the form Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 in two selectable modes solution only and step by step (or teach ) mode. There is also a link to his description of the solving methods. http://www.alpertron.com.ar/QUAD.HTM

63. Diophantine Equation Diophantine Equation Calculus Beyond discussion Given that x and y are positive integers such that 13x + 4y = 100 Then, what is x + y like? http://www.physicsforums.com/showthread.php?t=118531

64. 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

Diophantine m -tuples The problem of the construction of Diophantine m -tuples, i.e. sets with the property that product of any two of its distinct elements is one less then a square, has a very long history. There are some new results in this area, but many open problems and unproved conjectures still remains. In this survey we explain the main problems and results concerning Diophantine m -tuples. If you know something that you believe should be included, please let me know. Andrej Dujella Table of contents Introduction Diophantine quintuple conjecture Sets with the property D ... References (226 titles)

65. Diophantine Equation--2nd Powers -- From Wolfram MathWorld A general quadratic Diophantine equation in two variables x and y is given by ax^2+cy^2=k, (1) where a, c, and k are specified (positive or negative) integers and x and y are http://mathworld.wolfram.com/DiophantineEquation2ndPowers.html

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Piezas Diophantine Equation2nd Powers A general quadratic Diophantine equation in two variables and is given by where , and are specified (positive or negative) integers and and are unknown integers satisfying the equation whose values are sought. The slightly more general second-order equation is one of the principal topics in Gauss's Disquisitiones arithmeticae ) can be solved completely using solutions to the Pell equation . In particular, all solutions of are among the convergents of the continued fractions of the roots of Solution to the general bivariate quadratic Diophantine equation is implemented in Mathematica as Reduce eqn Element x y Integers x y For quadratic Diophantine equations in more than two variables, there exist additional deep results due to C. L. Siegel. An equation of the form where is an integer is a very special type of equation called a Pell equation . Pell equations, as well as the analogous equation with a minus sign on the right, can be solved by finding the continued fraction for . The more complicated equation can also be solved for certain values of and , but the procedure is more complicated (Chrystal 1961). However, if a single solution to (

66. Rational Triangles Triangles in the Euclidean plane such that all three sides are rational. With tables of Heronian and Pythagorean triples. http://grail.cba.csuohio.edu/~somos/rattri.html

Rational Triangles Definition Define a Rational Triangle as a triangle in the Euclidean plane such that all three sides measured relative to each other are rational. Once, it was thought that all triangles were rational. The discovery of counterexamples is attributed to the Pythagoreans. Any triangle similar to a rational triangle is rational also. Take as a unit the greatest common measure of the three sides. Then the length of the sides are positive integers whose greatest common measure is unity. All rational triangles can be uniquely constructed in this way from three positive integers with greatest common divisor unity and each less than the sum of the other two (triangle inequality). Right Triangles Define a Rational Right Triangle as a right triangle which is a rational triangle. A right triangle is a rational triangle if and only if all six trigonometrical ratios of the two complementary acute angles are rational. It can be proved, using the inscribed circle, that the tangent of its half angles are rational numbers. Define a Heronian Angle as an angle such that the tangent of its half is a rational number. Conversely, any rational number between zero and one is the tangent of half an acute angle of a rational right triangle. The rational number associated with one acute angle has numerator and denominator both odd, but for the other angle they are of different parity. All rational right triangles are constructed in this way from a rational number between zero and one in both ways.

67. 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://www.merriam-webster.com/dictionary/diophantine equation

68. Diophantine Equation Solutions Equations Problems General *are Diophantine Equation Solutions Equations Problems General *are Economy. http://www.economicexpert.com/a/Diophantine:equation.htm

69. Pell's Equation Record solutions. http://www.ieeta.pt/~tos/pell.html

Large fundamental solutions of Pell's equation Introduction Results References Links ... [Up] Introduction Let A be a positive integer which is not a perfect square. It is well known that there exist an infinite number of integer solutions of the equation Ax^2+1=y^2 , known as Pell's equation . The first non-trivial solution of this Diophantine equation, from which all others are easily computed, can be found using, e.g., the cyclic method , known in India in the 12th century, or using the slightly less efficient but more regular English method (17th century). There are other methods to compute this so-called fundamental solution , some of which are based on a continued fraction expansion of the square root of A Results This page describes our attempts to find the values of A that give rise to record-holders in the number of iterations of the cyclic and/or English methods, denoted respectively by n_c(A) and n_e(A) , or in the size of the fundamental solution, denoted by s(A) . We define the size of the fundamental solution as the number of base 10 digits of the smallest y higher than one that solves Pell's equation (actually, we use the base 10 logarithm of the fundamental solution). As usual

70. Math Forum Discussions - Comp.soft-sys.matlab This is just a linear Diophantine equation, or a system thereof. Oriole This is a Diophantine equation, as is the partitions problem, http://mathforum.org/kb/search!execute.jspa?forumID=80&q=diophantine&ran

71. Rational And Integral Points On Higher-dimensional Varieties Some of conjectures and open problems, compiled at AIM. http://aimath.org/WWN/qptsurface2/

Rational and integral points on higher dimensional varieties This web page highlights some of the conjectures and open problems concerning Rational and integral points on higher dimensional varieties. If you would like to print a hard copy of the whole outline, you can download a dvi postscript or pdf version. Lecture Notes Colliot-Thelene 1: Rational points on surfaces with a pencil ... Colliot-Thelene 2: Rational points on surfaces with a pencil ... de Jong: Rationally Connected Varieties ... Miscellaneous Photos The individual contributions may have problems because converting complicated TeX into a web page is not an exact science. The dvi, ps, or pdf versions are your best bet.

72. What Is The Simplest Form Of The Diophantine Equation? Who Solved The General Eq 1 post 1 author - Last post Jun 8What is the simplest form of the Diophantine equation Who solved the general equation for it and why was he not credited for the equation http://www.funtrivia.com/askft/Question81694.html

Fun Trivia Quizzes Games People ... Challenge List Welcome to our world of fun trivia quizzes and quiz games: New Player Play Now! New Questions Unanswered ... Post a Question Goto Qn # Archives What is the simplest form of the Diophantine equation? Who solved the general equation for it and why was he not credited for the equation? Question #81694. Asked by (Jun 08 07 3:54 PM) Triviaballer In most mathematical contexts, the linear form of an equation is the "simplest." Thus, I assume a Diophantine equation of the form "ax + by = 1" is the simplest. Brahmagupta (circa 628) appears to have solved their general forms, but he was not credited in the Western World because he methods were unknown at the time seemingly until Pierre de Fermat in the 1600s. http://en.wikipedia.org/wiki/Diophantine_equation#Hilbert.27s_tenth_problem Jun 08 07, 6:50 PM The responses above may be out of date. Create a Free ID instantly to see all recent responses , post your own follow-ups or questions, and access over trivia questions!

73. Diagonal Quartic Surfaces Articles, computations and software in Magma and GP by Martin Bright. http://www.boojum.org.uk/maths/quartic-surfaces/

Diagonal quartic surfaces This is Martin Bright's page about diagonal quartic surfaces. This page contains various bits and pieces which you can download. It is not really prepared for general use, and certainly not documented; but there is a small chance that somebody might want to use the programs and data presented here, so I have made them available. The programs, and the descriptions of them here, won't make much sense without reading the thesis. You can read my PhD thesis here ; some other articles are here Magma programs Here is a file which implements in Magma the algorithms for computing cohomology of the Picard group, both globally and locally. The algorithm for computing the Galois groups involved is available separately here , with more comments. GP scripts These are some scripts for the GP interpreter , which implement the various algorithms I wrote to deal with the Picard groups of diagonal quartic surfaces. nsgen.gp

74. Buscalo.com - La Web Diophantine equation Wikipedia, the free encyclopedia A linear Diophantine equation is an equation between two sums of monomials of degree zero . http://www.buscalo.com/index.php?page=search/web&search=diophantine equation

75. Pythagorean Triplets Pythagorean Triples A Javascript calculator for pythagorean triplets. http://www.hbmeyer.de/pythagen.htm

k = sequence m = n = x = k(n -m y = 2kmn z = k(n +m Pythagorean Triples A pythagorean triplet ("PT") consists of three natural numbers x, y and z with x + y = z PT's with greatest common divisor 1 ("TPT's") are of particular interest. Theorem 1 Every PT can be obtained in a unique way as a product of a TPT and a natural number k. Theorem 2 Theorem 3 x = n - m , y = 2mn , z = n + m resp. (z - x) = m (z + x) = n Theorem 4 to other interesting mathematical topics ... number theory interactive .. zur Seite in deutscher Sprache: Explanations for the "PT-calculator" (Java Script required) (resulting from a mathematics-workgroup at Faust-Gymnasium Staufen with Thorsten Jahrsetz and Jakob Weick, class 9d, 1999/2000): k = : the greatest common divisor k of a PT is edited here. sequence m = n = : input for m and n: button: "take m and n". The PT and the corresponding sequence will be shown. x = y = z = Buttons: A B C : edition of a sequence by mouseclick. Button: Reduce Theorem 5 Button: Reset : the original state will be restored. Much material about PT can be found at: http://mathworld.wolfram.com/PythagoreanTriple.htm

76. Introduction To Diophantine Equations To make sure you understand the technique we used to solve our Diophantine equation it would be a good exercise to solve the following equation by yourself before reading on 7x+18y=1 http://www.geometer.org/mathcircles/diophantine.pdf

77. Pythagorean Triples, Etc. A web text by Fred Barnes on 60-, 90-, and 120-degree integer-sided triangles. http://www.pythag.net

Next: Generating all Pythagorean Triples Contents Abstract: Observations on Pythagorean triples, on the solutions of 120 degree and 60 degree integer-sided triangles, and on deriving complex numbers from the law of cosines. Pythagorean Triples, etc. Fred Barnes fredlb@centurytel.net Generating all Pythagorean Triples Finding m and n if given a primitive Pythagorean triple Pythagorean triples and the divisors 3, 4, and 5. ... About this document ... fred 2010-05-08

78. Diophantine Equation (mathematics) -- Britannica Online Encyclopedia Diophantine equation (mathematics), equation involving only sums, products, and powers in which all the constants are integers and the only solutions of interest are integers. http://www.britannica.com/EBchecked/topic/164341/Diophantine-equation

document.write(''); Search Site: Advanced Search With all of these words With the exact phrase With any of these words Without these words Home SHOP BROWSE BLOG ... HELP Username: Password: Remember me Forgot your password? Help Email " is the e-mail address you used when you registered. Password " is case sensitive. If you need additional assistance, please contact customer support Enter the e-mail address you used when enrolling for Britannica Premium Service and we will e-mail your password to you. Diophantine-equation My Britannica CREATE MY Diophantine ... NEW ARTICLE ... SAVE Diophantine equation Table of Contents: Diophantine equation Article Article Related Articles Related Articles External Web sites External Web sites Citations Article Related Articles External Web sites Citations ARTICLE from the Diophantine equation equation involving only sums, products, and powers in which all the constants are integers and the only solutions of interest are integers. For example, 3 x y = 1 or x y z , where x y , and z are integers. Named in honour of the 3rd-century Greek mathematician

79. Index The official Beal Conjecture site with information and links regarding the problem. http://www.bealconjecture.com

The Beal Conjecture Background Mathematicians have long been intrigued by Pierre Fermat's famous assertion that A x + B x = C x is impossible (as stipulated) and the remark written in the margin of his book that he had a demonstration or "proof". This became known as Fermat's Last Theorem (FLT) despite the lack of a proof. Andrew Wiles proved the relationship in 1994, though everyone agrees that Fermat's proof could not possibly have been the proof discovered by Wiles. Number theorists remain divided when speculating over whether or not Fermat actually had a proof, or whether he was mistaken. This mystery remains unanswered though the prevailing wisdom is that Fermat was mistaken. This conclusion is based on the fact that thousands of mathematicians have cumulatively spent many millions of hours over the past 350 years searching unsuccessfully for such a proof. It is easy to see that if A x + B x = C x then either A, B, and C are co-prime or, if not co-prime that any common factor could be divided out of each term until the equation existed with co-prime bases. (Co-prime is synonymous with pairwise relatively prime and means that in a given set of numbers, no two of the numbers share a common factor.) You could then restate FLT by saying that A x + B x = C x is impossible with co-prime bases. (Yes, it is also impossible without co-prime bases, but non co-prime bases can only exist as a consequence of co-prime bases.)

80. Search Results For Rational-Solutions-of-the-Diophantine Rational Solutions of the Diophantine Equation y = x 2 . Diophantine equation - Wikipedia, the free encyclopedia http://www.homefitnessexerciseonline.com/Rational-Solutions-of-the-Diophantine-E

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