1. Developing A General 2nd Degree Diophantine Equation X2 + P = 2n Methods to solve these equations. http://www.biochem.okstate.edu/OAS/OJAS/thiendo.htm

Developing A General 2 nd Degree Diophantine Equation x + p = 2 n Thien Do Westmoore High School Science Department Oklahoma City, Oklahoma 73170 Abstract Introduction Methodology Conclusions ... References Abstract It is fun to experiment with numbers and exciting to discover patterns. Number theory played an important role in the Diophantine Equation. In this project, I consider a family of Diophantine equation: x + p = 2 n for various odd primes p. Using methods of congruences, I have shown that if p = 3 there is only one positive solution (1,2), and if p is any other odd prime not congruent to 7 mod 8, there are no solutions. The explanation of this general 2 nd degree equation’s solutions has not been previously determined as a result of the complication. This equation is solved uniquely by using congruences in modulo 2 and modulo 8. Introduction In the branch of number theory concerned with determining the solutions in integers of algebraic equations with two or more unknowns, Greek algebra and number theory played an important role in the appearance of the Arithmetica written by Diophantus. Diophantus was interested in exact solutions rather than the approximate solutions considered perfectly appropriate. Diophantus found interest in polynomial equation in one or more variables for which it is necessary to find a solution in either integers or rational numbers. This polynomial equation bears the name: Diophantine Equation Diophantus’s edition of the Arithmetica caught the attention of Pierre de Fermat (1601-1665), known as the “prince of amateur mathematician.” He discovered and developed many theorems in number theory. The most famous of Fermat’s assertion is the equation

2. 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

If you are using that software, you should enable JavaScript, and then reload this page.

3. PhD Thesis Thomas Stoll, TU Graz, 2003. Text (PS). http://finanz.math.tugraz.at/~stoll/thesis.htm

PhD Thesis Finiteness results for Diophantine equations for polynomial families (Stoll Thomas) ( ps We study the general Diophantine equation A P_m(x)+ B P_n(y)=C in integers x, y, where A, B, C are fixed rational numbers and P_m(x) PhD thesis advisor: Tichy Robert, Dr.phil., O.Univ.-Prof. organization: Working Group Mathematics A of the Institute of Mathematics year of publication: Last modified: November 22, 2003

4. 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:

5. Diophantine Equation. The American Heritage® Dictionary Of The English Language diophantine equation. The American Heritage® Dictionary of the English Language Fourth Edition. 2000. 2000. diophantine equation. NOUN An algebraic equation with two or more variables whose coefficients are integers http://www.bartleby.com/61/55/D0235550.html

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6. SOLVE A DIOPHANTINE EQUATION SOLVE A diophantine equation. In the fields below, enter the INTEGER coefficients of x and y and the click on the solve it button. Equation   x   +   y   =   http://www.math.csusb.edu/notes/maple/plot/dioph.html

Previous: Tools page Up: Contents page SOLVE A DIOPHANTINE EQUATION In the fields below, enter the INTEGER coefficients of x and y and the constant term. Then click on the solve it button. Peter Williams Sat Oct 26 23:31:28 PDT 1996

7. Diophantine Equation From FOLDOC diophantine equation. mathematics Equations with integer coefficients to which dinosaurs mating «. diode «. diophantine equation ». DIP ». diplex ». DirectAccess Storage Device http://wombat.doc.ic.ac.uk/foldoc/foldoc.cgi?Diophantine equation

8. Diophantine Equation -- From MathWorld diophantine equation. A diophantine equation is an equation in whichonly integer solutions are allowed. Hilbert s 10th problem asked http://mathworld.wolfram.com/DiophantineEquation.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Number Theory Diophantine Equations Diophantine Equation A Diophantine equation is an equation in which only integer solutions are allowed. Hilbert's 10th problem asked if an algorithm existed for determining whether an arbitrary Diophantine equation has a solution. Such an algorithm does exist for the solution of first-order Diophantine equations. However, the impossibility of obtaining a general solution was proven by Yuri Matiyasevich in 1970 (Matiyasevich 1970, Davis 1973, Davis and Hersh 1973, Davis 1982, Matiyasevich 1993) by showing that the relation (where is the th Fibonacci number ) is Diophantine. More specifically, Matiyasevich showed that there is a polynomial P in n m , and a number of other variables x y z , ... having the property that

9. Diophantine Equation From MathWorld diophantine equation from MathWorld A diophantine equation is an equation in which only integer solutions are allowed. Hilbert's 10th problem asked if an algorithm existed for determining http://rdre1.inktomi.com/click?u=http://mathworld.wolfram.com/DiophantineEquatio

10. Diophantine Equation--3rd Powers -- From MathWorld diophantine equation3rd Powers. As a part of last theorem was established.Thue showed that a diophantine equation of the form, (3). http://mathworld.wolfram.com/DiophantineEquation3rdPowers.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Number Theory Diophantine Equations Diophantine Equation3rd Powers As a part of the study of Waring's problem , it is known that every positive integer is a sum of no more than 9 positive cubes ( ), that every "sufficiently large" integer is a sum of no more than 7 positive cubes ( although it is not known if 7 can be reduced), and that every integer is a sum of at most 5 signed cubes ( although it is not known if 5 can be reduced to 4). It is known that every n can be written is the form An elliptic curve of the form for n an integer is known as a Mordell curve The 3.1.2 equation is a case of Fermat's last theorem with n = 3. In fact, this particular case was known not to have any solutions long before the general validity of Fermat's last theorem was established. Thue showed that a Diophantine equation

11. Diophantine Equation3rd Powers From MathWorld diophantine equation3rd Powers from MathWorld As a part of the study of Waring's problem, it is known that every positive integer is a sum of no more than 9 positive cubes (g(3)=9), that http://rdre1.inktomi.com/click?u=http://mathworld.wolfram.com/DiophantineEquatio

12. 11D: Diophantine Equations (Thus the diophantine equation x^2+y^2=N can be treated both in 11P and herein 11D (as a Pell equation).). I. diophantine equations , Nieuw Arch. Wisk. 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.

13. HOME PAGE PIETER MOREE Pieter Moree, University of Amsterdam On my PhD thesis. My PhD thesisis entitled On the psixyology of diophantine equations . I http://web.inter.nl.net/hcc/J.Moree/linkind2.htm

Pieter Moree, University of Amsterdam On my PhD thesis My PhD thesis is entitled 'On the psixyology of Diophantine equations'. I defended it September first, 1993 in Leiden. It was written under the direction of Prof. R. Tijdeman. In Chapter 1, entitled 'General introduction' I give a general introduction to psixyology and formulate the main results contained in my thesis. In Chapter 2, 'Introduction to psixyology' I give a general introduction and survey of psixyology. In Chapter 3, In Chapter 4 'Abstract psixyology' I generalize this result to arithmetic semigroups and give various examples. In Chapter 5, In Chapter 6, `Some Ramanujan-Nagell equations with many solutions' In Chapter 7, `On arithmetical progressions having only few different prime factors in comparison with their length' , I resolve a conjecture due to Shorey and Tijdeman on finite arithmetic progressions having only few different prime factors. This chapter has appeared, considerably extended and completely rewritten as [Ch7]. Both include the following table.

14. Wakabayashi's Home Page Seikei University, Tokyo. diophantine equations and transcendence problems for values of analytic functions. http://www.ge.seikei.ac.jp/wakaba/

Žá—ÑŒ÷‚̃z[ƒ€ƒy[ƒW‚ւ悤‚±‚» Wakabayashi Isao's Home Page English ZŠF§180-8633 @@@“Œ‹ž“s•‘ –ìŽs‹gËŽ›–k’¬@3-3-1 @@@¬æü‘åŠwHŠw•” Address: Department of Engineering, Seikei University, @@ Kichijoji Kitamachi 3, Musashino-shi, Tokyo 180-8633, Japan E-mail: wakaba@ge.seikei.ac.jp Tel.@: 0422-37-3808 (Office) ‚±‚̍ Žv‚¤‚±‚Æ ‚킽‚µ‚Ì’S“–‰È–Ú my research

15. MSN Encarta - Dictionary - Diophantine Equation dionysian. Dionysus. diophantine equation. diopside. diopter. dioptric No thesaurus result for " diophantine equation" http://encarta.msn.com/encnet/features/dictionary/DictionaryResults.aspx?refid=1

16. LINEAR DIOPHANTINE EQUATIONS A web tool for solving diophantine equations of the form ax + by = c. http://thoralf2.uwaterloo.ca/htdocs/linear.html

Solving ax +by = c a b c

17. Diophantine Equation - Wikipedia, The Free Encyclopedia diophantine equation. From Wikipedia, the free encyclopedia. A traditionalname for the study of diophantine equations is Diophantine analysis. http://en.wikipedia.org/wiki/Diophantine_equation

Diophantine equation From Wikipedia, the free encyclopedia. In mathematics 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. A traditional name for the study of Diophantine equations is Diophantine analysis . The questions asked include: Are there any solutions? Are there any solutions beyond some that are easily found by inspection? Are there finitely or infinitely many solutions? Can all solutions be found, in theory? Can one in practice compute a full list of solutions? Such problems often lay unsolved for centuries, and mathematicians gradually came to understand their depth (in some cases), rather than treat them as puzzles. In 1970, a novel result in mathematical logic known as Matiyasevich's theorem showed that it is hopeless to expect a complete theory, in effect settling Hilbert's tenth problem . The point of view of Diophantine geometry , which is the application of algebraic geometry techniques in this field, has continued to grow as a result; since treating

18. Merriam-Webster Online One entry found for diophantine equation. Main Entry Di·o·phan·tine equation 10 Search Results for "Diophantine+equation" For More Information on "Diophantine+equation" go to http://www.m-w.com/cgi-bin/dictionary?book=Dictionary&va=Diophantine equatio

19. Solving General Pell Equations John Robertson's treatise on how to solve diophantine equations of the form x^2 dy^2 = N. http://hometown.aol.com/jpr2718/pelleqns.html

Main Math htmlAdWH('7002737', '234', '60'); Solving the generalized Pell equation x - Dy =N John Robertson An improved version of what used to be here is now a PDF file at my homepage . Look for the PDF file titled ``Solving the generalized Pell equation.'' The old HTML page (uncorrected, un-enhanced, on some browsers some math symbols do not display correctly) is at old HTML page. This page is best viewed using Microsoft Internet Explorer (MS IE). Last Modified August 3, 2002 John P. Robertson JPR2718@AOL.COM This page has been visited times.

20. DiophantineEquation diophantine equation (English). Search for diophantine equation inNRICH PLUS maths.org Google. Definition level 2. An equation http://thesaurus.maths.org/dictionary/map/word/967

Home Preferences Help User Guides ... Forum thesaurus.maths.org alphabetical galleries topics For Quick Reference: drag this m-button to your links toolbar Find word or phrase containing matching starting with ending with in English English Danish German Spanish ... Slovak Diophantine equation (English) Search for " Diophantine equation " in NRICH PLUS maths.org Google Definition level 2 An equation in which the coefficients are integers, and the solutions are also required to be integers. These equations often look deceptively simple and require difficult number theory to solve them. Show graph Requires Java Relations broader: (en) Equation narrower: (en) Linear Diophantine equation (en) Pell's equation references: (en) Number theory referenced: (en) Pythagorean triple Funded by: EU Socrates Minerva, HeyMath!, Cambridge University Press

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