Geometry.Net - Math_Discover: Diophantine Equation

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
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
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
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

Extractions: POINTERS: Texts Software Web links Selected topics here 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.
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

Extractions: 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.
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

Extractions: 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. 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:
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

Extractions: Back to Table of Contents Biographies of Mathematicians - Diophantus Born: about 200 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. 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).
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

Extractions: Back to Table of Contents Biographies of Mathematicians - Diophantus Born: about 200 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. "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).
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
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
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/

Extractions: 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.
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/

Extractions: 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.
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

Extractions: 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, Bowmans 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. Julias 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.
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
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

Extractions: (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
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

Extractions: 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
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

Extractions: 30-Apr-1995 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. 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
The Prime Glossary: Logarithmic Function
logarithmic function (another Prime Pages' Glossary entries). The Prime Glossary.
http://primes.utm.edu/glossary/page.php?sort=Log
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

61-79 of 79 Back | 1 | 2 | 3 | 4