61. AoPS Math Forum :: View Topic - Diophantine Equation The time now is Fri Aug 05, 2005 806 am All times are GMT 7 Hours, Diophantineequation, Mark the topic unread Math Forum » Olympiad Section » Number http://www.artofproblemsolving.com/Forum/topic-46449.html

The AoPS Intermediate Trigonometry/Complex Numbers course begins Thursday, Sept 22. Enrollment deadline is October 2. Font Size: The time now is Fri Sep 16, 2005 3:38 pm All times are GMT - 7 Hours Diophantine equation Mark the topic unread Math Forum Olympiad Section ... Number Theory Unsolved Problems Moderators: amfulger Arne harazi Megus ... pbornsztein Page of [6 Posts] Printer View Tell a Friend View previous topic View next topic Author Message alishkov New Member Joined: 31 Jul 2005 Posts: 8 Posted: Mon Aug 01, 2005 12:20 am Post subject: Diophantine equation The problem is actually from the Russian Math Olympiad this year. Using very advanced number theory, I managed to prove that the following equations do not have integer solutions. Unfortunately, I would truly appreciate an elegant or elementary solution to the problem: Show that the equations 2x^2+1=y^15 and 2x^2-1=y^15 do not have non-trivial solutions. I tried factoring the expressions y^15+-1 but the problems reduce to Pelle-like equations which should have common solutions. After that I couldn't do anything more. I hope someone manages to help me solve these problems.

62. AoPS Math Forum :: View Topic - Easy Diophantine Equation All times are GMT 7 Hours, Easy diophantine equation 2005 659 pm Postsubject Easy diophantine equation, 1 Mark this post and the followings unread http://www.artofproblemsolving.com/Forum/topic-46028.html

The AoPS Intermediate Trigonometry/Complex Numbers course begins Thursday, Sept 22. Enrollment deadline is October 2. Font Size: The time now is Fri Sep 16, 2005 3:38 pm All times are GMT - 7 Hours Easy diophantine equation Mark the topic unread Math Forum Olympiad Section ... Number Theory Moderators: amfulger Arne harazi Megus ... pbornsztein Page of [6 Posts] Printer View Tell a Friend View previous topic View next topic Author Message Jutaro P versus NP Joined: 10 Apr 2005 Posts: 32 Posted: Thu Jul 28, 2005 6:59 pm Post subject: Easy diophantine equation Source: VII Centroamerican and Caribbean Olympiad 2005, Problem 2 Show that the equation has no integer solutions. This problem is due to Arnoldo Aguilar (El Salvador) No hay tiempo que perder Enfermera de sombras y distancias Yo vuelvo a ti huyendo del reino incalculable De ¡ngeles prohibidos por el amanecer Back to top Soarer Navier-Stokes Equations Joined: 30 Aug 2003 Posts: 1573 Posted: Fri Jul 29, 2005 2:13 am Post subject: is a perfect square. We only need to check the 8k+3, 4, 7, factors of 2002.

63. Applicatons - Solve Diophantine Equations Instructions This function will enable one to find all integer solutions to theabove equation, also known as a diophantine equation. http://archives.math.utk.edu/articles/atuyl/confrac/diaph.html

Solve Diophantine Equations A * X + B * Y = C where a,b, and c are given, non-zero integers Instructions: This function will enable one to find all integer solutions to the above equation, also known as a Diophantine equation In the space provided below enter the integer values of A,B, and C. When this is completed, press the bar below. The value of A: The value of B: The value of C: Main Introduction History Applications ... Credits

64. Continued Fractions - Introduction Equations of the first form are called diophantine equations. Those of the secondform are named Pell s equations. Definition 10 http://archives.math.utk.edu/articles/atuyl/confrac/intro.html

What is a continued fraction? To introduce this web site, the most appropriate place to start is with a definition of a continued fraction. A continued fraction refers to all expressions of the form where a ,a ,a ,.... and b ,b ,b ,... are either real or complex values. The number of terms can be either finite or infinite. What can I find at this site on continued fractions? Let me answer this question by first explaining what you will not find at this site. You will not find any deep analysis of continued fractions. For those of you researching this area, I direct your attention to the resources Even though this site does not go into great analysis of continued fractions, it does cover some of the basic theorems of the field. For those of you who are "math-phobic," now would probably be a good time to leave. When I created this site, my intention was to provide a brief introduction to this fascinating area of mathematics. I have aimed this presentation at those at my own level, that is, those at an undergraduate level with an interest in mathematics. For many, this may be a first introduction to continued fractions since this subject, if it is taught at all, is restricted to a single chapter in a number theory text. Hopefully this site will inspire others to study continued fractions in greater detail. I have broken this site into four main pieces. In the first section, which is found below, I present some of the basic theorems about simple continued fractions. In the second

65. Mathematical Programming Glossary Page 2 diophantine equations. A system of equations whose variables must be In general, solving the linear diophantine equations, Ax=b for x in Zn, http://carbon.cudenver.edu/~hgreenbe/glossary/second.php?page=D.html

66. Science: Mathematics: Number Theory: Diophantine Equations - Open Site Science Mathematics Number Theory diophantine equations Open Site. http://open-site.org/Science/Mathematics/Number_Theory/Diophantine_Equations/

Open Site The Open Encyclopedia Project home submit content become an editor the entire directory only in Number_Theory/Diophantine_Equations Top Science Mathematics Number Theory : Diophantine Equations Fermat's Last Theorem The assertion that the equation x^n + y^n = z^n has no solutions in non-zero integers x,y,z when n > 2. The equation was originally proposed by Fermat in the margin of his copy of the works of Diophantus in 1605, who claimed to have a proof which, famously, 'the margin was too small to contain'. It was proved by Andrew Wiles in a proof announced in 1993 and completed with Richard Taylor in 1994. The proof deduces the result as a consequence of the proof of the deeper Shimura-Taniyama-Weil conjecture on elliptic curves over the rational numbers. Attempts to prove Fermat's Last Theorem in the 19th century led to the development of much of modern algebraic number theory. Pell's Equation The equation x^2 - d y^2 = 1 for given d. It is soluble in integers x,y for all positive non-square d. The fundamental solution is that with the smallest non-zero value of y. All other solution can be obtained from the fundamental.

67. Diophantine Equation From FOLDOC diophantine equation. mathematics Equations with integer coefficients to whichinteger solutions are sought. Because the results are restricted to http://foldoc.doc.ic.ac.uk/foldoc/foldoc.cgi?Diophantine equation

68. MATLAB Central File Exchange - Generate Many Examples Of Ramanujam's Diophantine x y and u v and N that satisfy the diophantine equation x^3 + y^3 = u^3 +v^3 = N where the four integers x, y, u, v have no common factor. http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=5870&ob

69. Gcd (MATLAB Functions) These are useful for solving diophantine equations and computing In the nextexample, we solve for x and y in the diophantine equation 30x + 56y = 8 . http://www.mathworks.com/access/helpdesk/help/techdoc/ref/gcd.html

MATLAB Function Reference gcd Greatest common divisor Syntax G = gcd(A,B) [G,C,D] = gcd(A,B) Description G = gcd(A,B) returns an array containing the greatest common divisors of the corresponding elements of integer arrays A and B . By convention, gcd(0,0) returns a value of ; all other inputs return positive integers for G [G,C,D] = gcd(A,B) returns both the greatest common divisor array G , and the arrays C and D , which satisfy the equation: A(i).*C(i) + B(i).*D(i) = G(i) . These are useful for solving Diophantine equations and computing elementary Hermite transformations. Examples The first example involves elementary Hermite transformations. For any two integers a and b there is a -by- matrix E with integer entries and determinant = (a unimodular matrix) such that: E * [a;b] = [g,0] where g is the greatest common divisor of a and b as returned by the command [g,c,d] = gcd(a,b). The matrix E equals: c d -b/g a/g In the case where a = 2 and b = 4 [g,c,d] = gcd(2,4) g = c = d = So that E = In the next example, we solve for x and y in the Diophantine equation [g,c,d] = gcd(30,56)

70. Diophantine Equations In Naturals -- References Solving linear diophantine equations using the geometric structure of the the basis of nonnegative solutions to a linear diophantine equation. http://www.ncc.up.pt/~apt/dioph/hdioph_1.html

References References Ana Paula, On Solving Linear Diophantine Constraints . Doctoral thesis, Universidade do Porto, 1997. and Filgueiras, 1997a] Proceedings of 8th Conference on Rewriting Techniques and Applications RTA'97 , Lecture Notes in Computer Science 1232, 269-283, Springer-Verlag,1997. Reviewed in Math Reviews and Filgueiras, 1997b] Progress in Artificial Intelligence EPIA'97 , Lecture Notes in Artificial Intelligence 1323, 73-84, Springer-Verlag, 1997. Reviewed in , 884.11020; (Reviewer J. Piehler) 11D04, 90C10. [Filgueiras Journal of Symbolic Computation , 19, 507-526, 1995. Reviewed in Math Reviews [Filgueiras Progress in Artificial Intelligence 6th Portuguese Conference on Artificial Intelligence , Lecture Notes in Artificial Intelligence 727, 297-306, Springer-Verlag, 1993. Reviewed in Math Reviews and Filgueiras, 1991a] Proceedings of the 5th Portuguese Conference on Artificial Intelligence , Lecture Notes in Artificial Intelligence 541, Springer-Verlag, 30-44, 1991. and Filgueiras, 1997c] COMPULOG NET Workshop on Constraint Programming and Filgueiras, 1993]

71. Linear Diophantine Equations Linear diophantine equations are linear equations in which only integer solutionsare allowed. Consider a system of $m$ equations in $n$ variables for which http://www.irisa.fr/polylib/DOC/node33.html

Next: Smith decomposition Up: Other tools Previous: Other tools Contents Linear Diophantine equations Linear Diophantine equations are linear equations in which only integer solutions are allowed. Consider a system of equations in variables for which we look for integral solutions. is a matrix and is a vector of order In the homogeneous space, the equation is where To solve such a sytems, first the rows of are rearranged in such a way that the first rows of are the ones which contribute to the rank. This is done with: static void RearrangeMatforSolveDio (Matrix *M) : rearrange the matrix in order to solve a diofantine equation. Then the function SolveDiophantine for solving the equation can be used. If a solution exists, the procedure returns , otherwise it returns int SolveDiophantine (Matrix *M, Matrix **U, Vector **X) : solve Diophantine Equations Generally this functions is used in connection with operations on lattices because a lattice can be seen as a solution of a Diophantine equation. Next: Smith decomposition Up: Other tools Previous: Other tools Contents Sorin Olaru 2002-04-24

72. Diophantine Equation diophantine equation by Annie (Nov 30, 2004). Re diophantine equation by mars (Dec1, 2004) Re Re diophantine equation by mars (Dec 2, 2004) http://at.yorku.ca/cgi-bin/bbqa?forum=ask_an_algebraist_2004;task=show_msg;msg=0

73. AMCA: On The Diophantine Equation $p^x-q^y=c$ By Florian Luca The diophantine equation pxqy=c in positive integers (p, q, x, y, c) with p andq distinct primes has received considerable interest. http://at.yorku.ca/c/a/k/l/14.htm

Atlas Mathematical Conference Abstracts Conferences Abstracts Organizers ... About AMCA Journées Arithmétiques XXIII July 6-12, 2003 University of Graz and University of Technology of Graz Graz, Styria, Austria Organizers S. Frisch, A. Geroldinger, P. Grabner, F. Halter-Koch, C. Heuberger, G. Lettl, R. Tichy View Abstracts Conference Homepage On the diophantine equation p x -q y =c by Florian Luca UNAM The diophantine equation p x -q y =c in positive integers (p, q, x, y, c) with p and q distinct primes has received considerable interest. What is usually of interest for this equation is the following question: Given p and q, find all the values of c for which the above equation has at least two distinct solutions (x, y). The existence of two solutions (x, y) for the above equation reduces to the existence of a nontrivial solution of the equation p x -q y =p x -q y in positive integers (x , y , x , y ), where by nontrivial we mean that (x , y ) =/= (x , y In my talk, I will present two results concerning equation (1). I will first show that if one of the primes p and q, say p, is fixed, then there exist only finitely many quintuples of positive integers (q, x , y , x , y ) with q =/= p and prime which give a nontrivial solution of (1). I will then show that the ABC-conjecture implies that there should exist only finitely many sixtuples of positive integers (p, q, x

74. Section 2.3: Linear Diophantine Equations 1 2.3 Linear diophantine equations 1. As stated in the Prelab section, one goal ofthis chapter is to develop a systematic method for finding all integer http://www.math.mtu.edu/mathlab/COURSES/holt/dnt/euclid3.html

2.3 Linear Diophantine Equations 1 As stated in the Prelab section, one goal of this chapter is to develop a systematic method for finding all integer solutions x and y (when there are any) to the linear diophantine equation a x b y c where a b , and c are integer constants. As you discovered when working on the Prelab exercises, for a given choice of a and b , equation (1) may have several solutions or possibly no solutions; it depends on the value of c . If we have specific values for a and b , how can we figure out the values of c for which equation (1) will have a solution? As a first step, let's look at the situation for a specific choice of a and b , say a = 6 and b = 4. In this case, the above equation becomes x y = c and our question is: For what values of c is there a solution to this equation? One way to approach this is to try plugging a bunch of different values for x and y into the left-hand side of (2), and see what we get out. After all, any value that comes out must be a suitable value for

75. Section 2.5: Linear Diophantine Equations 2 2.5 Linear diophantine equations 2. Now we re getting somewhere. With the researchquestions in the preceding section complete, we now know all of the http://www.math.mtu.edu/mathlab/COURSES/holt/dnt/euclid5.html

2.5 Linear Diophantine Equations 2 Now we're getting somewhere. With the research questions in the preceding section complete, we now know all of the solutions to the equation a x b y d where d = gcd( a b ). Therefore, it remains to determine all of the solutions to the equation a x b y c where c kd and k is an integer. To get a feel for what is going on, let's look at an example. In the preceding section, we looked at the equation 7 x y = 1. Let's change this a bit, say to x y Using the Euclidean Algorithm, we found that x y x y = 1. Thus, it is easy to see that x y x y = 5. What about the other solutions? The applet from the previous section can be used to search for additional solutions: Your browser does not support java. Research Question 5 Suppose that gcd( a b d and that ( x y ) is a solution to a x b y d . Find the general form of all solutions ( x y ) to a x b y k d giving x in terms of x and y in terms of y Section 2.1 Section 2.2 Section 2.3 Section 2.4 ... DNT Table of Contents

76. LMS JCM (8) 116-121 Motivated by this point of view, an example of a diophantine equation The diophantine equation has four terms rather than the usual three terms http://www.lms.ac.uk/jcm/8/lms2004-056/

The LMS JCM Published 18 May 2005. First received 13 Oct 2004. A diophantine equation associated to X Imin Chen Abstract: Several classes of Fermat-type diophantine equations have been successfully resolved using the method of galois representations and modularity. In each case, it is possible to view the proper solutions to the diophantine equation in question as corresponding to suitably defined integral points on a modular curve of level divisible by 2 or 3. Motivated by this point of view, an example of a diophantine equation associated to the modular curve X0(5) is discussed in this paper. The diophantine equation has four terms rather than the usual three terms characteristic of generalized Fermat equations. This paper is available as (94 KB) here for details. Go to the Volume 8 index Return to the LMS JCM Homepage

77. Science Math Number Theory Diophantine Equations - Science Math Number Theory Di Science Math Number Theory diophantine equations , web directory and searchengine, featuring a directory of millions of links along with thumbnails of http://www.abc.net/directory/Science/Math/Number_Theory/Diophantine_Equations/

HOME DIRECTORY META SEARCH NEWS ... World Search: search the entire directory search this category only Top Science Math Equal Sums of Like Powers ... Fermat's Last Theorem This category in other languages: Spanish Bibliography on Hilbert's Tenth Problem - Searchable, ~400 items. 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 - Statement of the problem in several languages, history of the problem, bibliography and links to related WWW sites. 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.

78. The Diophantine Equation P(x) = N! And A Result Of M. Overholt 2000 Mathematics Subject Classification. 11D85. Key words and phrases. ABCConjecture,diophantine equations. Glasnik Matematicki Home Page. http://www.math.hr/glasnik/vol_37/no2_04.html

Glasnik Matematicki, Vol. 37, No.2 (2002), 269-273. THE DIOPHANTINE EQUATION P x n ! AND A RESULT OF M. OVERHOLT Florian Luca Mathematical Institute, UNAM, Ap. Postal 61-3 (Xangari), CP 58089, Morelia, Michoacan, Mexico e-mail: fluca@matmor.unam.mx Abstract. In this note, we show that the ABC-conjecture implies that a diophantine equation of the form P x n ! with P a polynomial with integer coefficients and degree d > 1 has only finitely many integer solutions ( x n ) with n 2000 Mathematics Subject Classification. Key words and phrases. ABC-Conjecture, diophantine equations. Glasnik Matematicki Home Page

79. Solving The Diophantine Equation Ax^2+by^2=m By Cornacchia's Method Solving the diophantine equation ax2+by2=m by Cornacchia s method. Here a 0,b 0,m a+b, gcd(a,m)=1=gcd(a,b). The algorithm is from A. Nitaj, http://www.numbertheory.org/php/cornacchia.html

Solving the diophantine equation ax +by =m by Cornacchia's method The algorithm is from A. Nitaj, L'algorithme de Cornacchia , Expositiones Mathematicae 13 (1995), 358-365. We find the positive solutions (x,y) with gcd(x,y)=1. This works for m with up to say 20 digits, due to the limitations of the program used to factor m. Last modified 20th January 2005 Return to main page

80. Gödel's Theorem For example, 11x538xy8+18133=0 is a diophantine equation. Solving Diophantineequations - ie finding their solutions or showing that they have no http://www.sm.luth.se/~torkel/eget/godel/self.html

Is self-reference essential to Gödel's theorem? 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.

Page 4     61-80 of 92    Back | 1  | 2  | 3  | 4  | 5  | Next 20