41. Diophantine Equations A diophantine equation is one in which the solutions must be integers or rationalnumbers. Before turning to the formal theory of diophantine equations, http://www25.brinkster.com/ranmath/diophan/

var google_language="en"; var adHB=true; wDoL("top","RR1WOKH"); wCls("RR1WOKH"); wDoL("btm","RR1WOKH"); showA("RR1WOKH"); Web Hosting by Brinkster Diophantine Equations A Diophantine equation is one in which the solutions must be integers or rational numbers. On this page we shall restrict ourselves to solutions in positive integers. Let us imagine one's experience on his first encounter with the following puzzle: There's Dad and Ma and Brother and Me; The sum of our ages is eighty-three. Six times Dad's age is seven times Ma's And she is three times Me. How old is Dad? The ordinary algebra student says, "Hey, I can't solve these equations because there are four unknowns and only three equations." Indeed there is no unique solution over the field of the real numbers, but there is a unique solution in integers. Here is another problem of the same sort: At Jimmy's school four-fifths of the students use five-sixths of the desks. What is the smallest possible number of students in the class? Take these problems to school, bring them up in math class, and watch Teacher slowly turn into a babbling idiot.

42. Diophantine Equations A similar algorithm solves this problem for linear diophantine equations with any Second order diophantine equations. The next step would be to consider http://www25.brinkster.com/ranmath/diophan/diophan.htm

var google_language="en"; var adHB=true; wDoL("top","NX72UQN"); wCls("NX72UQN"); wDoL("btm","NX72UQN"); showA("NX72UQN"); Web Hosting by Brinkster Diophantine Equations The following material is mostly quoted from a web page created by Dr. Karlis Podnieks of the University of Latvia. The original is (or was) here Linear Diophantine equations Problems that can be solved by finding solutions of algebraic equations in the domain of integer numbers have been 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: x = (17y+1)/7 = 2y + (3y+1)/7. The number (3y+1)/7 must be integer, let us denote it by z.Then 3y+1=7z and x=2y+z. Thus we have arrived at the equation3y-7z=-1 having smaller coefficients than the initial one. Let us apply our "coefficient reduction method" once more: y = (7z-1)/3 = 2z + (z-1)/3.

43. SOLVE A DIOPHANTINE EQUATION SOLVE A diophantine equation enter the INTEGER coefficients of x and y andthe constant term. Then click on the solve it button. Equation 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

44. Linear Diophantine Equations Linear diophantine equations. Linear diophantine equations. Diophantus, aGreek mathematician who lived during the 4th century AD, was one of the first http://web.usna.navy.mil/~wdj/book/node7.html

Next: Binary and -ary notation Up: Divisibility Previous: The greatest common divisor Contents Index Linear diophantine equations Diophantus, a Greek mathematician who lived during the 4th century A.D., was one of the first people who attempted to find integral or rational solutions to a given system of equations. Often the system involves more unknowns than equations. We will consider a linear equation, , with two unknowns Theorem 1.2.17 The linear equation has no solutions if does not divide . If does divide then there are infinitely many solutions given by: where is any solution and is any integer. proof : The first part of the theorem follows from lemma . Let be any solution, and let be any other solution. We want to show that and , where . Substitute into the equation: Therefore, . If we can divide both sides of this equation by Since (see the Exercise ), it follows that . Substituting into the above equation gives: and our proof is complete. Corollary 1.2.18 If then proof : Assume . Let . By the above theorem, there exist integers

45. Diophantine Equation - Definition Of Diophantine Equation In Encyclopedia In mathematics, a diophantine equation is an equation between two polynomialswith integer coefficients with any number of unknowns. http://encyclopedia.laborlawtalk.com/Diophantine_equation

Add to Favorites General Encyclopedia Legal ... Law forum Search Word: Visit our Law forums Labor Law Accidents Law Bankruptcy Law Business Law ... Tax Law In mathematics , a Diophantine equation is an equation between two polynomials with integer coefficients with any number of unknowns. A Diophantine problem is given as a Diophantine equation, whose solutions are the possible assignments of integers for the unknowns for which the equation is satisfied. The word Diophantine refers to the Greek mathematician of the third century A.D., Diophantus of Alexandria , who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra . The mathematical study of Diophantine problems Diophantus initiated is now called Diophantine analysis A linear Diophantine equation is an equation between two sums of monomials of degree zero or one. Contents showTocToggle("show","hide") 1 Examples of Diophantine equations 2 Diophantine analysis 2.1 Traditional questions 2.2 Hilbert's tenth problem ... 3 External links Examples of Diophantine equations ax by = 1: See Bézout's identity ; this is a linear Diophantine.

46. BBC - H2g2 - Diophantine Equations h2g2 is the unconventional guide to life, the universe and everything, a guidethat s written by visitors to the website, creating an organic and evolving http://www.bbc.co.uk/dna/h2g2/A964956

@import url('/includes/tbenh.css') ; Home TV Radio Talk ... A-Z Index Friday 16th September 2005 Text only Guide ID: A964956 (Edited) Edited Guide Entry SEARCH h2g2 Advanced Search New visitors: Returning members: BBC Homepage The Guide to Life The Universe and Everything 3. Everything Mathematics Created: 11th March 2003 Diophantine Equations Front Page What is h2g2? Who's Online Write an Entry ... Help Like this page? Send it to a friend! Most of us have encountered simple equations such as 5x + 3y = 10. We know (or we may have a vague recollection from our school days) that if there are two unknown values (or 'variables'), you need two equations to be able to get a unique answer for x and y. With only one equation, absolutely any value of x can be used and a corresponding value of y can be calculated, which is not normally very useful. Diophantine equations are different. They are named after the Greek mathematician, Diophantus of Alexandria, who lived in the 3rd century and developed many of the principles of number theory. In this sort of equation, there is a restriction that all the numbers involved must be whole numbers without any fractional or decimal part. That is, they must have an integer value. Finding values of x and y that fit these conditions can be much more difficult, but as a result is usually more useful. Consider the following problem...

47. Number Theory: Diophantine Equations A diophantine equation is an algebraic equation in one or more unknowns with The simplest case is the linear diophantine equation in two unknowns http://uzweb.uz.ac.zw/science/maths/zimaths/62/dioph.htm

Number Theory: Diophantine Equations Introduction 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. 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: which incidently shows that the greatest common divisor (840, 611) = 1. From these equations, we have derived the following expressions: Combining these operations, we obtain the development of the rational number [ 840/611] in the form An expression of the form a = a a a :+ [ 1/(a n + [ 1/(a n where the a's are positive integers, is called a

48. Miscellaneous Diophantine Equations Miscellaneous diophantine equations. x2 + y2 = z4, x + y = w2. In a letter to St.Martin and Frenicle on May 31, 1643, Fermat suggested the problem of http://www.mathpages.com/home/kmath022/kmath022.htm

Miscellaneous Diophantine Equations x + y = z ,   x + y = w In a letter to St. Martin and Frenicle on May 31, 1643, Fermat suggested the problem of finding a Pythagorean triple x,y,z such that x+y and x were both squares.  (Actually, he asked for a rational right triangle whose hypotenuse and the sum of whose legs are squares.)  He then gave the solution and stated that it was the smallest possible.  Dickson describes several solution methods, including the one used by Fermat and another method by Euler, who noted that the problem is equivalent to finding two integers x,y such that  x + y is a square and x + y is a fourth power.  First he noted that if we set x = p q and y = 2pq then x + y is equal to the square (p + q .  Then we can make p + q   a square by setting p = r s and q = 2rs, which gives x + y = (r + s .  This also gives Our objective now is to find rational numbers r,s such that the above expression yields a square.  It’s easy to find a quadratic in r and s whose square matches the first, second, and last terms, so the difference between this square and the above expression will be of the form Ars (Br+Cs), which vanishes if we set r = C and s = -B.  In this case (r

49. Diophantine Equations diophantine equations. diophantine equations. Many problems in number theoryare concerned with finding integral or rational solutions of systems of http://www.math.okstate.edu/~wrightd/4713/nt_essay/node2.html

Next: Pythagorean Triples Up: No Title Previous: Typical Problems in Number Diophantine Equations Many problems in number theory are concerned with finding integral or rational solutions of systems of polynomial equations. This kind of problem was first described by Diophantus, a Greek mathematician living in the third century A.D. Pythagorean Triples Fermat's Last Theorem Mordell's Equation Waring's Problem ... The Congruent Number Problem David J. Wright

50. Section 1.1 From "Hilbert's Tenth Problem" By Yuri MATIYASEVICH Let us recall that a diophantine equation is an equation of the form In addition to (1.1.1), diophantine equations can be written in the more general http://logic.pdmi.ras.ru/~yumat/H10Pbook/par_1_1.htm

Section 1.1 from the book "HILBERT's TENTH PROBLEM" written by Yuri MATIYASEVICH 1.1 Diophantine equations as a decision problem Let us recall that a Diophantine equation is an equation of the form D x x m where D is a polynomial with integer coefficients. In addition to (1.1.1), Diophantine equations can be written in the more general form D L x x m D R x x m where D L and D R are again polynomials with integer coefficients. When we speak of ''an arbitrary Diophantine equation,'' we shall have in mind an equation of the form (1.1.1) since an equation of the form (1.1.2) can easily be transformed into an equation of the form (1.1.1) by transposing all the terms to the left-hand side. However, we will often use the notation (1.1.2) for particular equations if this form turns out to be easier to grasp. We shall also take another advantage of the more general form (1.1.2); namely, in this case we can demand that D L and D R are polynomials with non-negative coefficients. Diophantine equations typically have several unknowns, and we must distinguish the degree of with respect to a given unknown x i and the ( total degree of (1.1.1), i.e., the maximum, over all the monomials constituting the polynomial

51. Open Problem? Diophantine Equations. Open problem given a diophantine equation that has a solution for every n ega+b+c =abc , compute the expected number of steps until a solution is http://semillon.wpi.edu/~aofa/AofA/msg00031.html

ANALYSIS of ALGORITHMS Bulletin Board Date Prev Date Next Thread Prev Thread Next ... Thread Index Open problem? Diophantine equations. To : "Marko Riedel" < Subject : Open problem? Diophantine equations. From : "Marko Riedel" < Date : Fri, 27 Feb 2004 15:16:41 +0100 http://www.geocities.com/markoriedelde/index.html http://www.geocities.com/markoriedelde/index.html Follow-Ups Open problem? Diophantine equations. , Marko Riedel, 03/12/2004 References A challenge for hispanophones. , Marko Riedel, 02/11/2004 Prev by Date: A challenge for hispanophones. Next by Date: Re: Una verdadera maravilla. (Sister Celine.) Previous by thread: A challenge for hispanophones. Next by thread: Open problem? Diophantine equations. Index(es): Date Thread

52. Open Problem? Diophantine Equations. diophantine equations. From Marko Riedel mriedel@xxxxxxxxxxxxx ; Date Fri,12 Mar 2004 145955 + diophantine equations., Marko Riedel, 02/27/2004 http://semillon.wpi.edu/~aofa/AofA/msg00033.html

ANALYSIS of ALGORITHMS Bulletin Board Date Prev Date Next Thread Prev Thread Next ... Thread Index Open problem? Diophantine equations. To : "Marko Riedel" < Subject : Open problem? Diophantine equations. From : "Marko Riedel" < Date : Fri, 12 Mar 2004 14:59:55 +0100 Hi folks, I spent a week learning how to manipulate hypercomplex numbers in order to compute certain combinatorial sums and wrote a Maple library for this purpose. It's on my combinatorics page on my home page. http://www.geocities.com/markoriedelde/combnumth.html http://www.geocities.com/markoriedelde/index.html References A challenge for hispanophones. , Marko Riedel, 02/11/2004 Open problem? Diophantine equations. , Marko Riedel, 02/27/2004 Prev by Date: Re: Una verdadera maravilla. (Sister Celine.) Next by Date: Hypercomplex numbers and combinatorial sums. (II) Previous by thread: Open problem? Diophantine equations. Next by thread: Hypercomplex numbers and combinatorial sums. (II) Index(es): Date Thread

53. Hilbert's Tenth Problem. Diophantine Equations. Part 2. By K.Podnieks What is Mathematics? Goedel s Theorem and Around. Textbook for students. Section 4.Part 2. By K.Podnieks. http://www.ltn.lv/~podnieks/gt4a.html

Back to title page Left Adjust your browser window Right 4. Hilbert's Tenth Problem 4.1. History of the Problem. Story of the Solution 4.2. Plan of the Proof 4.3. Investigation of Fermat's Equation We will investigate only a special (the simplest!) case of Fermat's equation - where D=a x - (a -1)y No problems to prove the existence of non-trivial solutions for this equation: you can simply take x=a, y=1. After this, all the other natural solutions we can calculate by using the following smart idea. Let us note that x - (a -1)y = (x+y*sqrt(a -1)) * (x-y*sqrt(a Take our first non-trivial solution x=a, y=1: a - (a -1) = (a+sqrt(a -1)) * (a-sqrt(a Consider the n-th power: (a+sqrt(a n * (a-sqrt(a n Now let us apply the Newton's binomial formula to the expression (a+sqrt(a n . For example, if n=2, then   (a+sqrt(a = a + 2a*sqrt(a -1) + (a I.e. some of the items contain sqrt(a -1), and some do not. Let us sum up either kind of the items: (a+sqrt(a n = x n (a) + y n (a)sqrt(a where x n (a), y n (a) are natural numbers. For example, x (a)=2a -1, y

54. 3. The Problem Of Simplest Diophantine Representation 3. The Problem of Simplest Diophantine Representation. Next, I take thecomplexity, or simplicity, of a diophantine equation to be the number of basic http://www.hf.uio.no/filosofi/njpl/vol2no2/diophantine/node3.html

Next: 4. On Classical and Up: The Problem of the Previous: 2. Diophantine Sets and 3. The Problem of Simplest Diophantine Representation Let me now turn to the proper subject of this paper. To begin with, I assume that a formalized language of arithmetic , that has a finite stock of basic symbols, has been fixed. Obviously, this is assumed to include the standard logical symbols, a constant symbol (e.g. ` ') for 0, the successor symbol (e.g. ` S ' or ` '), and two function symbols (e.g. ` x ' and `+') for addition and multiplication. Next, I take the complexity , or simplicity , of a Diophantine equation to be the number of basic symbols occurring in it. Now recall that the modern ``inverted'' approach to the subject begins with a set of ``solutions'' and attempts to find a corresponding equation. Now it is indeed a very short and natural step to add that given a set of ``solutions'' one would like to find a maximally simple equation. And this will be our problem. To make it exactly defined, let us formulate it as follows: ``Given a finite set of numbers S , what is the simplest (in terms of the number of basic symbols it contains) Diophantine equation P x x x n ), such that

55. 3. Diophantine Equations And Logic I believe that Skolem considered diophantine equations and logic as one subject . The theory of diophantine equations considers questions about http://www.hf.uio.no/filosofi/njpl/vol1no2/pioneer/node3.html

Next: 4. Introducing new symbols Up: Thoralf Skolem: Pioneer of Previous: 2. Vita 3. Diophantine equations and logic I believe that Skolem considered Diophantine equations and logic as one subject. Let me indicate a little speculatively how Skolem's thought might have been. The theory of Diophantine equations considers questions about satisfiability of equations like This is really not far from questions about satisfiability of logical statements. A statement can be seen as a ``Diophantine equation'' where we have In addition, we have to indicate how the quantifiers are to be interpreted. This is of course more complicated. So Skolem's first work was to see how the quantifiers could be interpreted. Nordic Journal of Philosophical Logic, Vol. 1, No. 2, pp. 107117.

56. Diophantine Equation Exponentialtime algorithm for solution of Diophantine linear equation. http://www.geocities.com/SiliconValley/Garage/3323/aat/a_diop.html

Diophantine linear equation PROBLEM A.1,...,A.N T ALGORITHM My Diophant algorithm solves almost always diophantine equation for and IMPLEMENTATION Unit: internal subroutine Global variables: array A.1,...,A.N of positive integers Parameters: a positive integer N , a positive integer T Result: displays in the screen the solution of the problem - i. e. a subset A. whose sum is equal T . The execution is halted (via the exit statement) as soon as a solution is found Interface: internal procedure QUICKSORT DIOPHANT: procedure expose A. parse arg N, T call QUICKSORT N Ls.1 = A.1 do I = 2 to N Im1 = I - 1; Ls.I = A.I + Ls.Im1 end S = 1; Stack.1 = N T parse var Stack.S R T V; S = S - 1 if Ls.K = T then call EXIST V, K, 1 if A.R = T then call EXIST V, R, R D = V A.L; S = S + 1 Stack.S = (L - 1) (T - A.L) D end end end say "Solution not exist" exit EXIST: procedure expose A. parse arg V, B, E do J = B to E by -1; V = V A.J; end say "Solution:" V exit COMPARISON For N=100;T=25557 and the array A. created by statements: Seed = RANDOM(1, 1, 481989)

57. Diophantine Equation --  Encyclopædia Britannica diophantine equation equation involving only sums, products, and powers in whichall the constants are integers and the only solutions of interest are http://www.britannica.com/eb/article-9030556

Home Browse Newsletters Store ... Subscribe Already a member? Log in Content Related to this Topic This Article's Table of Contents Diophantine equation Print this Table of Contents Shopping Price: USD $1495 Revised, updated, and still unrivaled. The Official Scrabble Players Dictionary (Hardcover) Price: USD $15.95 The Scrabble player's bible on sale! Save 30%. Merriam-Webster's Collegiate Dictionary Price: USD $19.95 Save big on America's best-selling dictionary. Discounted 38%! More Britannica products Diophantine equation Page 1 of 1 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 Diophantus of Alexandria , these equations were first systematically solved by Hindu mathematicians beginning with Aryabhata I Diophantine equation...

58. Diophantine Equations Find It Science Math Number Theory diophantine equations MAGMA code tosolve diophantine equations of the form F(x)=G(y), for which Runge s http://www.ebroadcast.com.au/dir/Science/Math/Number_Theory/Diophantine_Equation

SEARCH GUIDE NEWS AUSTRALIAN TV GUIDE DVD RENTALS ... Number Theory : Diophantine Equations Science The A to Z of science is right here. Equal Sums of Like Powers Fermat's Last Theorem Bibliography on Hilbert's Tenth Problem Searchable, ~400 items. Developing A General 2nd Degree Diophantine Equation x^2 + p = 2^n Methods to solve these equations. Diagonal Quartic Surfaces Articles, computations and software in Magma and GP by Martin Bright. Diophantine Equations Dave Rusin's guide to Diophantine equations. Diophantine Geometry in Characteristic p A survey by Jos© Felipe Voloch. 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. Diophantus Quadraticus On-line Pell Equation solver by Michael Zuker. Egyptian Fractions Lots of information about Egyptian fractions collected by David Eppstein. The Erdos-Strauss Conjecture Fermat's Method of Infinite Descent Notes by Jamie Bailey and Brian Oberg. Illustrates the method on FLT with exponent 4. Hilbert's Tenth Problem 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.

59. Diophantine Equation -- Facts, Info, And Encyclopedia Article A linear diophantine equation is an equation between two sums of (Click link for The depth of the study of general diophantine equations is shown by the http://www.absoluteastronomy.com/encyclopedia/d/di/diophantine_equation.htm

Diophantine equation [Categories: Diophantine equations] In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics , a Diophantine equation is a (A mathematical expression that is the sum of a number of terms) polynomial (A mathematical statement that two expressions are equal) equation that only allows the variables to be (Any of the natural numbers (positive or negative) or zero) integer s. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. The word Diophantine refers to the (A native or inhabitant of Greece) Greek mathematician of the third century A.D., (Greek mathematician who was the first to try to develop an algebraic notation (3rd century)) Diophantus of (The chief port of Egypt; located on the western edge of the Nile delta on the Mediterranean Sea; founded by Alexander the Great; the capital of ancient Egypt) Alexandria , who made a study of such equations and was one of the first mathematicians to introduce (The practice of investing things with symbolic meaning) symbolism into (The mathematics of generalized arithmetical operations) algebra . The mathematical study of Diophantine problems Diophantus initiated is now called Diophantine analysis. A linear Diophantine equation is an equation between two sums of (Click link for more info and facts about monomials) monomials of degree zero or one.

60. Diophantine Equation - YourDictionary.com - American Heritage Dictionary Search Mamma.com for diophantine equation diophantine equation n. An algebraicequation with two or more variables whose coefficients are integers, http://www.yourdictionary.com/ahd/d/d0235550.html

Search Mamma.com for "Diophantine equation" Search: Normal Definitions Short defs (Pronunciation Key) Diophantine equation n. An algebraic equation with two or more variables whose coefficients are integers, studied to determine all integral solutions. [After Diophantus , third-century A.D. Greek mathematician.] Back to Search Back The American Heritage Dictionary of the English Language, Fourth Edition

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