41. A Public-key Cryptosystem Based On Diophantine Equation IPSJ JOURNAL Abstract Vol.31 No.12 016. A Public-key Cryptosystem Based on DiophantineEquation. YAGISAWA MASAHIRO ?1. ?1 Showa Engineering Corporation http://www.ipsj.or.jp/members/Journal/Eng/3112/article016.html

Last Update¡§Thu May 24 14:42:10 2001 IPSJ JOURNAL Abstract Vol.31 No.12 - 016 A Public-key Cryptosystem Based on Diophantine Equation YAGISAWA MASAHIRO Showa Engineering Corporation ¢¬Index Vol.31 No.12 IPSJ Journal Contents Web Members Service Menu Comments are welcome. Mail to address editt@ips j.or.jp , please.

42. Mathematica Information Center: The Linear Diophantine Equation In Nonnegative V The Linear diophantine equation in Nonnegative Variables, Mathematica code is usedto solve the linear diophantine equation for nonnegative integer values. http://library.wolfram.com/database/Articles/3176/

All Collections Articles Books Conference Proceedings Courseware Demos MathSource: Packages and Programs Technical Notes Title The Linear Diophantine Equation in Nonnegative Variables Author H. Greenberg Journal / Anthology Mathematica in Education and Research Year: Volume: Issue: Page range: Description Mathematica code is used to solve the linear diophantine equation for nonnegative integer values. Skiena's Mathematica implementation of Dijkstra's shortest path algorithm is used for the problem solutions. Subject Mathematics Number Theory Downloads Greenberg.nb (35.2 KB) - Mathematica Notebook

43. ?(Diophantine Equation) ? ? The summary for this Korean page contains characters that cannot be correctly displayed in this language/character set. http://gifted.kaist.ac.kr:7777/math/a2007_001h01.html

44. About The Diophantine Equation Y(Y+m1)(Y+m2)(Y+ M1+m2)=2X(X+m1)(X+m2)(X+ M1+m2) Title, About the diophantine equation Y(Y+m1)(Y+m2)(Y+ m1+m2)=2X(X+m1)(X+m2)(X+m1+m2). Date, 200101-01. Subject, Natural Sciences Mathematics. http://arc.cs.odu.edu:8080/dp9/getrecord/oai_dc/oai:hofprints:hofprints00000026

OAI Header Identifier oai:hofprints:hofprints00000026 Datestamp Dublin Core Metadata Creator Akbik, Safwan Description Let m be the greatest common factor of the two positive integers m1 and m2. In this paper we show that if m has a "specific form," then the nontrivial solutions of the equation of the title are m times the "primitive solutions" of a similar equation with smaller m1 and m2. Also it is shown that the equation of the title with m1=1 and m2=3 has only four pairs of nontrivial solutions in integers given by X=14 or *18, and Y=17 or *21. Then we shall find all solutions in integers of the equation of the title if m2=3m1 and m1 is of a specific form. Title About the Diophantine Equation Y(Y+m1)(Y+m2)(Y+ m1+m2)=2X(X+m1)(X+m2)(X+ m1+m2) Date Subject Natural Sciences: Mathematics Identifier http://hofprints.hofstra.edu/documents/disk0/00/00/00/26/index.html Type Preprint Link to other metadata formats

45. [FWP] Diophantine Equation Solver (of Much Interest) Thread Index FWP diophantine equation Solver (of much interest). SubjectFWP diophantine equation Solver (of much interest); http://bumppo.net/lists/fun-with-perl/2000/06/msg00068.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index [FWP] Diophantine Equation Solver (of much interest) To fwp@technofile.org perl5-porters@perl.org Subject [FWP] Diophantine Equation Solver (of much interest) From jeffp@crusoe.net Date: Fri, 23 Jun 2000 10:52:04 -0400 (EDT) cc: wendigo@pobox.com http://www.pobox.com/~japhy/regexes/diophantine I'll be commenting the code tomorrow (I'm leaving yapc now, and will not be able to comment it until I'm home). Thanks for your time. Jeff "japhy" Pinyan japhy@pobox.com http://www.pobox.com/~japhy/ PerlMonth - An Online Perl Magazine http://www.perlmonth.com/ The Perl Archive - Articles, Forums, etc. http://www.perlarchive.com/ CPAN - #1 Perl Resource (my id: PINYAN) http://search.cpan.org/ Prev by Date: Re: [FWP] REALLY spiffy typeglob code, but with a significant weakness Next by Date: [FWP] algorithm to calculate pi Prev by thread: RE: Re: [FWP] algorithm to calculate pi Next by thread: [FWP] last minute fun: review of YAPC presentation Navigation: Date Index Thread Index Search Other lists at bumppo.net

46. Mathematics Encyclopedia -- Platonic Realms Platonic Realms Home http://www.mathacademy.com/cgi-bin/ref_main.cgi?Diophantine equation

47. A Binomial Diophantine Equation a question of Richard K. Guy in proving that 21 choose 2 = 10 choose 4 is the largestsolution of the binomial diophantine equation n choose 2 = m choose 4. http://netec.mcc.ac.uk/WoPEc/data/Papers/dgreureir1997103.html

mirrored in Providing the latest research results since 1993 Search tips: title=fiscal or author=levine Working Papers Series Journals Authors JEL Classification ... Econometric Institute Report >> A binomial diophantine equation A binomial diophantine equation Weger, B.M.M. de (Econometric Institute) Econometric Institute Report 103 / Erasmus University Rotterdam, Econometric Institute ( web site (RePEc:dgr:eureir:1997103) Abstract: We answer a question of Richard K. Guy in proving that 21 choose 2 = 10 choose 4 is the largest solution of the binomial diophantine equation n choose 2 = m choose 4. Creation: 1994 Keywords: Normale verdeling go top Information for authors: Are you an author of this paper? Please take the time and register at our new service. Read all about it at: http://netec.mcc.ac.uk/HoPEc/geminiabout.html . Note you do not need to register in order to use the search service!! Download (Main Text) See this paper's references Access statistics WoPEc is a RePEc service managed by Jose Manuel Barrueco and Thomas Krichel contact us Last updated: 2003-03-16 05:07:39

48. One Diophantine Equation Abstract We solve a problem posed recently by JHE Cohn, in proving that x = y =1 is the only solution in nonnegative integers to the diophantine equation x^2 http://netec.mcc.ac.uk/WoPEc/data/Papers/dgreureir19972.html

mirrored in Providing the latest research results since 1993 Search tips: title=fiscal or author=levine Working Papers Series Journals Authors JEL Classification ... Econometric Institute Report >> One diophantine equation One diophantine equation Weger, Benjamin M.M. de (Econometric Institute, Erasmus University Rotterdam) Econometric Institute Report 2 / Erasmus University Rotterdam, Econometric Institute ( web site (RePEc:dgr:eureir:19972) Abstract: We solve a problem posed recently by J.H.E. Cohn, in proving that x = y = 1 is the only solution in nonnegative integers to the diophantine equation x^2 - 3 y^4 = -2. Creation: 1995 go top Information for authors: Are you an author of this paper? Please take the time and register at our new service. Read all about it at: http://netec.mcc.ac.uk/HoPEc/geminiabout.html . Note you do not need to register in order to use the search service!! Download (Main Text) See this paper's references Access statistics WoPEc is a RePEc service managed by Jose Manuel Barrueco and Thomas Krichel contact us Last updated: 2003-03-16 05:07:39

49. Cryptanalysis Of A Diophantine Equation Oriented Public Key Cryptosystem pp. 511512 Cryptanalysis of a diophantine equationOriented Public Key Cryptosystem. http://www.computer.org/tc/tc1997/t0511abs.htm

p p. 511-512 Cryptanalysis of a Diophantine Equation Oriented Public Key Cryptosystem C.s.  Laih, M.j.  Gau Abstract —Lin et al. proposed a new public-key cipher system whose security is based upon the Diophantine equations. In this note, we show that their scheme is insecure under the ciphertext only attack. Index Terms- ... Public keys, knapsack cryptosystem, cryptography, cryptanalysis. The full text of IEEE Transactions on Computers is available to members of the IEEE Computer Society who have an online subscription and an web account

50. Maths Glossary diophantine equation An equation which is required to be solved by integers.Thusthe diophantine equation 2x 2 3y 2 = 5 has the integer solution x=2,y=1 http://www.fortunecity.com/emachines/e11/86/mathglos.html

web hosting domain names email addresses related sites Maths Glossary Algorithm : A fixed process which if caried out systematically produces a desired result.Thus the "Euclidean algorithm" is a set of rules which when applied to two integers produces their common divisor. Complex Number : A 2 dimensional number comprising a real and imaginary component of the form A+Bi,where i is the square root of -1.Such numbers are mapped on the Argand plane and form a matrix of numbers,the imaginary numbers being at right angles to the real ones.The conjugate takes the form a-bi. Continuum Hypothesis : In Cantorian set theory,the cardinal number of a set designates its "manyness". The cardinality of the set of integers 1,2,3,... is designated by .The cardinality of the set of real numbers is 2 .The continuum hyopothesis asserts that there is no set whose cardinal falls between and 2 Cube : 1. A number raised to the power 3,ie 2 x 2 x 2 = 2 2. A 3 dimensional figure with the height,width and depth having the same lengths,and all at right-angles to each other.

51. Historical Notes: Hilbert's Tenth Problem posed the problem of finding a single finite procedure that could systematicallydetermine whether a solution exists to any specified diophantine equation. http://www.wolframscience.com/reference/notes/1161a

From: Stephen Wolfram, A New Kind of Science Notes for Chapter 12: The Principle of Computational Equivalence Section: Implications for Mathematics and Its Foundations Page 1161 Hilbert's tenth problem. Beginning in antiquity various procedures were developed for solving particular kinds of Diophantine equations (see page 1171). In 1900, as one of his list of 23 important mathematical problems, David Hilbert posed the problem of finding a single finite procedure that could systematically determine whether a solution exists to any specified Diophantine equation. The original proof of Gödel’s Theorem from 1931 in effect involves showing that certain logical and other operations can be represented by Diophantine equations - and in the end Gödel’s Theorem can be viewed as saying that certain statements about Diophantine equations are unprovable. The notion that there might be universal Diophantine equations for which Hilbert’s Tenth Problem would be fundamentally unsolvable emerged in work by Martin Davis in 1953. And by 1961 Davis, Hilary Putnam and Julia Robinson had established that there are exponential Diophantine equations that are universal. Extending this to show that Hilbert’s original problem about ordinary polynomial Diophantine equations is unsolvable required proving that exponentiation can be represented by a Diophantine equation, and this was finally done by

52. Historical Notes: Diophantine Equations Particularly from the work of Carl Friedrich Gauss around 1800 there emergeda procedure to find solutions to any quadratic diophantine equation in two http://www.wolframscience.com/reference/notes/1164b

From: Stephen Wolfram, A New Kind of Science Notes for Chapter 12: The Principle of Computational Equivalence Section: Implications for Mathematics and Its Foundations Page 1164 Diophantine equations. If variables appear only linearly, then it is possible to use ExtendedGCD (see page 946) to find all solutions to any system of Diophantine equations - or to show that none exist. Particularly from the work of Carl Friedrich Gauss around 1800 there emerged a procedure to find solutions to any quadratic Diophantine equation in two variables - in effect by reduction to the Pell equation x^2==a y^2+1 (see page 947), and then computing ContinuedFraction[Sqrt[a]]. The minimal solutions can be large; the largest ones for successive coefficient sizes are given below. (With size s coefficients it is for example known that the solutions must always be less than (14 s)^(5s).). There is a fairly complete theory of homogeneous quadratic Diophantine equations with three variables, and on the basis of results from the early and mid-1900s a finite procedure should in principle be able to handle quadratic Diophantine equations with any number of variables. (The same is not true of simultaneous quadratic Diophantine equations, and indeed with a vector x of just a few variables, a system m . x^2 == a of such equations could quite possibly show undecidability.) Ever since antiquity there have been an increasing number of scattered results about Diophantine equations involving higher powers. In 1909

53. Diophantine Equation Exploration Exploration of a Diophantine Type Equation. A diophantine equation is any polynominalequation (in one or more unknowns) whose coefficients are integers. http://web.pdx.edu/~carbonec/assignment6.html

Exploration of a Diophantine Type Equation A diophantine equation is any polynominal equation (in one or more unknowns) whose coefficients are integers. The solutions of a diophantine equation are the integer values of the unknowns which satisfy the equation. This exploration will explore all real number solutions to one particular diophantine equation, whose integer coefficients are one. If you would like to read more about diophantine equations, you may click here for an excellent web site. This paper is an exploration of a particular group of diophantine equations of the form: x n +y n Exploration when n is even Exploration when n is odd Conclusion All figures were done in Nu Calc Created by Cheryl Carbone Student Math 588 Portland State Mathematics Department

54. MathGroup Archive: September 2001 Diophantine Equation Y^2=x^3-2*x+1 diophantine equation y^2=x^32*x+1. To mathgroup@smc.vnet.net; Subjectmg30666 diophantine equation y^2=x^3-2*x+1; From Ranko http://forums.wolfram.com/mathgroup/archive/2001/Sep/msg00027.html

January February March April ... Author Index Diophantine equation y^2=x^3-2*x+1 To mathgroup@smc.vnet.net Subject : [mg30666] Diophantine equation y^2=x^3-2*x+1 From bojanic@math.ohio-state.edu Date : Mon, 3 Sep 2001 20:32:40 -0400 (EDT) Sender owner-wri-mathgroup@wolfram.com Prev by Date: Diophantine equation y^2=x^3-2*x+1 Next by Date: Sticky Previews? Mathematica Export, Illustrator, and Textures Prev by thread: Diophantine equation y^2=x^3-2*x+1 Next by thread: Sticky Previews? Mathematica Export, Illustrator, and Textures

55. MathGroup Archive: September 2001 Diophantine Equation Y^2=x^3-2*x+1 diophantine equation y^2=x^32*x+1. To mathgroup@smc.vnet.net; Subjectmg30672 diophantine equation y^2=x^3-2*x+1; From Ranko http://forums.wolfram.com/mathgroup/archive/2001/Sep/msg00026.html

January February March April ... Author Index Diophantine equation y^2=x^3-2*x+1 To mathgroup@smc.vnet.net Subject : [mg30672] Diophantine equation y^2=x^3-2*x+1 From bojanic@math.ohio-state.edu Date : Mon, 3 Sep 2001 20:32:50 -0400 (EDT) Sender owner-wri-mathgroup@wolfram.com Prev by Date: Re: Fitting data to line with a specific slope Next by Date: Diophantine equation y^2=x^3-2*x+1 Prev by thread: Re:convert graphics Next by thread: Diophantine equation y^2=x^3-2*x+1

56. 3. The Problem Of Simplest Diophantine Representation Next, I take the complexity, or simplicity, of a diophantine equationto be the number of basic symbols occurring in it. 3 Now recall 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

57. 2. Diophantine Sets And Hilbert's Tenth Problem One can get rid of the negative coefficients by transforming a diophantine equationof the form P( ) = 0 to the form P'( ) = P''( ). Moreover, there are well http://www.hf.uio.no/filosofi/njpl/vol2no2/diophantine/node2.html

Next: 3. The Problem of Up: The Problem of the Previous: 1. Introduction 2. Diophantine Sets and Hilbert's Tenth Problem Let us first review the basic concepts and results related to the subject of Diophantine equations (cf. Davis 1973 Smorynski 1991 Matiyasevich 1993 First, note that traditionally both positive and negative integers have been allowed as solutions, and as coefficients. The modern approach, however, concentrates more on the natural number coefficients and solutions. But the difference is logically inessential. One can get rid of the negative coefficients by transforming a Diophantine equation of the form P ) = to the form P' P'' ). Moreover, there are well-known techniques for reducing the decision problem of the integer case to the non-negative one, and vice versa (see e.g. Davis 1973 Smorynski 1991 Matiyasevich 1993 ). In what follows, I shall therefore be somewhat indifferent between these two cases; everything that I say below applies equally to both theories of integers and theories of natural numbers. Accordingly, I shall use

58. Home Page Of Nikos Tzanakis 1 The diophantine equation x^3 + 3y^3 = 2^n , J.Number Th. 15 (1982),376387. 2 On the diophantine equation y^2 - D = 2^k , J.Number Th. http://itia.math.uch.gr/~tzanakis/

Nikolaos G. Tzanakis Professor of Mathematics Department of Mathematics University of Crete 714 09 IraklionCrete GREECE Office: H 303 Tel and fax: +30-2810-232962 (home), +30-2810-393839 (office) Home Address: 8 Solomou str., GR-713 06 IraklionCrete, GREECE E-mail: tzanakis@math.uch.gr Research Interests Explicit solution of diophantine equations I am extremely interested in LaTeX. Here is a nice example of a greek LaTeX document. If you are interested in downloading a greek-latin LaTeX, together with Ghostview and the WinShell (La)TeX editor, click here Research Publications The diophantine equation x^ n J.Number Th On the diophantine equation y^ - D = 2^ k J.Number Th ] (In cooperation with A.Bremner) Integer points on y^ = x^ Math.Comp The diophantine equation x^ - y^ = 1 and related equations J .Number Th The complete solution in integers of x^ n J. Number Th On the diophantine equation 2x^3 + 1 = py^2 Manuscr. Math A remark on a theorem of W.E.H.Berwick Math. Comp On the diophantine equation x^2 - Dy^4 = k Acta Arithm ] (In cooperation with J.Wolfskill)

59. Mathematical Publications By Benjamin M.M. De Weger (MR 92a11028, Zbl. 738.11029); A diophantine equation of Antoniadis, inRA Mollin (ed.), Number theory and applications, NATO ASI Series C, Vol. http://www.xs4all.nl/~deweger/publikaties.html

Mathematical publications by Benjamin M.M. de Weger Home Book Papers Educational Books ... Lecture Notes Book Algorithms for diophantine equations CWI-Tract no. 65, Centre for Mathematics and Computer Science, Amsterdam [1989]. (MR 90m:11025, Zbl. 687.10013) Originally appeared as Ph.D. Dissertation, University of Leiden [1987]. Top of this page Papers Approximation lattices of p-adic numbers, Journal of Number Theory (MR 87k:11069, Zbl. 595.10027) (with Products of prime powers in binary recurrence sequences, Part I: The hyperbolic case, with an application to the generalized Ramanujan-Nagell equation, Mathematics of Computation (MR 87m:11027a, Zbl. 623.10011) Products of prime powers in binary recurrence sequences, Part II: The elliptic case, with an application to a mixed quadratic-exponential equation, Mathematics of Computation (MR 87m:11027b, Zbl. 623.10012) Solving exponential diophantine equations using lattice basis reduction algorithms, Journal of Number Theory (MR 88k:11097, Zbl. 625.10013) Erratum

60. Section 1.1 From "Hilbert's Tenth Problem" By Yuri MATIYASEVICH 1.1 diophantine equations as a decision problem. Let us recall that a Diophantineequation is an equation of the form. D(x 1 , ,x m )=0,, (1.1.1). 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

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