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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

Extractions: 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. 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.

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

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

Extractions: 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 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 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

Extractions: Select Search All Bartleby.com All Reference Columbia Encyclopedia World History Encyclopedia Cultural Literacy World Factbook Columbia Gazetteer American Heritage Coll. Dictionary Roget's Thesauri Roget's II: Thesaurus Roget's Int'l Thesaurus Quotations Bartlett's Quotations Columbia Quotations Simpson's Quotations Respectfully Quoted English Usage Modern Usage American English Fowler's King's English Strunk's Style Mencken's Language Cambridge History The King James Bible Oxford Shakespeare Gray's Anatomy Farmer's Cookbook Post's Etiquette Bulfinch's Mythology Frazer's Golden Bough All Verse Anthologies Dickinson, E. Eliot, T.S. Frost, R. Hopkins, G.M. Keats, J. Lawrence, D.H. Masters, E.L. Sandburg, C. Sassoon, S. Whitman, W. Wordsworth, W. Yeats, W.B. All Nonfiction Harvard Classics American Essays Einstein's Relativity Grant, U.S. Roosevelt, T. Wells's History Presidential Inaugurals All Fiction Shelf of Fiction Ghost Stories Short Stories Shaw, G.B. Stein, G. Stevenson, R.L. Wells, H.G. Reference American Heritage Dictionary Diophantine analysis ... BIBLIOGRAPHIC RECORD The American Heritage Dictionary of the English Language: Fourth Edition.

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

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

Extractions: 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

Extractions: 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

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

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.

Pieter Moree, University of Amsterdam On my PhD thesis. My PhD thesisis entitled On the psixyology of diophantine equations . I

Extractions: 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.

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

Extractions: áÑ÷Ìz[y[WÖæ¤±» Wakabayashi Isao's Home Page English ZF§180-8633 @@@s ìsgËk¬@3-3-1 @@@¬æüåwHw 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) ±Ì v¤±Æ í½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

17. Diophantine Equation - Wikipedia, The Free Encyclopedia
http://en.wikipedia.org/wiki/Diophantine_equation

Extractions: 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: 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

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

Extractions: Find word or phrase containing matching starting with ending with in English English Danish German Spanish ... Slovak Search for " Diophantine equation " in NRICH PLUS maths.org Google 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

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