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  1. A note on the solvability of the diophantine equation: 1n [superscript n] + 2n [superscript n] + ... mn [superscript n] = G(m+1)n [superscript n] (Afdeling zuivere wiskunde) by J. van de Lune, 1975
  2. On the diophantine equation: Ap[x]+bq[y]=c+dp[z]q[w] by Christopher Skinner, 1989
  3. Diophantine equations in division algebras, by Ralph G Archibald, 1927
  4. On the Diophantine equation 1[superscript k] - 2[superscript k] -...- x[superscript k] - R(x) = y[superscript z] (Afdeling zuivere wickunde ; ZW 113/78) by Marc Voorhoeve, 1978
  5. Exponential Diophantine Equations (Cambridge Tracts in Mathematics) by T. N. Shorey, R. Tijdeman, 1987-02-27
  6. New methods for solving quadratic diophantine equations (part I and part II) (Research report) by A. G Schaake, 1989
  7. Certain quaternary quadratic forms and diophantine equations by generalized quaternion algebras by Lois Wilfred Griffiths, 1927
  8. Two-way counter machines and diophantine equations (Technical report / State University of New York at Buffalo, Department of Computer Science) by Eitan M Gurari, 1980
  9. On pairs of diophantine equations by Amin Abdul K Muwafi, 1959
  10. Diophantine equations: Lectures given by W.J. Ellison, 1971-1972 by William John Ellison, 1972
  11. Diophantine equations, with special reference to elliptic curves by J. W. S Cassels, 1966
  12. Diophantine Approximation and Diophantine Equations by Wolfgang M. Schmidt, 1990
  13. Notes on the Diophantine equation y²-k=x³ (Arkiv för matematik) by Ove Hemer, 1954
  14. Tables of solutions of the diophantine equation Y3 - X2 =: K by Mohan Lal, 1965

41. Diophantine Equation--Linear
diophantine equation encyclopedia article about Diophantine encyclopedia article about diophantine equation. diophantine equation in Freeonline English dictionary, thesaurus and encyclopedia. diophantine equation.
Diophantine EquationLinear
A linear Diophantine equation (in two variables) is an equation of the general form
where solutions are sought with , and Integers . Such equations can be solved completely, and the first known solution was constructed by Brahmagupta. Consider the equation
Now use a variation of the Euclidean Algorithm , letting and
Starting from the bottom gives
Continue this procedure all the way back to the top.
Take as an example the equation
Proceed as follows.
The solution is therefore . The above procedure can be simplified by noting that the two left-most columns are offset by one entry and alternate signs, as they must since
so the Coefficients of and are the same and
Repeating the above example using this information therefore gives and we recover the above solution. Call the solutions to and . If the signs in front of or are Negative , then solve the above equation and take the signs of the solutions from the following table: equation In fact, the solution to the equation is equivalent to finding the Continued Fraction for , with and Relatively Prime (Olds 1963). If there are

42. Diophantine Equations
diophantine equations. A diophantine equation is one in which thesolutions must be integers or rational numbers. On this page we
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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. Before turning to the formal theory of Diophantine equations, let us see what a little common sense will do.

43. Diophantine Equations
Thus we have a simple general method allowing to determine, given an arbitrarylinear diophantine equation with two unknowns, has it solutions in integer
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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.

44. Linear Diophantine Equations
Previous The greatest common divisor Contents Index Linear diophantineequations. Diophantus, a Greek mathematician who lived during
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. Randomness In Arithmetic
The mathematical assertion that the diophantine equation with parameter k has nosolution encodes the assertion that the kth computer program never halts.
Randomness in Arithmetic
Scientific American 259, No. 1 (July 1988), pp. 80-85
by Gregory J. Chaitin
It is impossible to prove whether each member of a family of algebraic equations has a finite or an infinite number of solutions: the answers vary randomly and therefore elude mathematical reasoning. What could be more certain than the fact that 2 plus 2 equals 4? Since the time of the ancient Greeks mathematicians have believed there is little-if anything-as unequivocal as a proved theorem. In fact, mathematical statements that can be proved true have often been regarded as a more solid foundation for a system of thought than any maxim about morals or even physical objects. The 17th-century German mathematician and philosopher Gottfried Wilhelm Leibniz even envisioned a ``calculus'' of reasoning such that all disputes could one day be settled with the words ``Gentlemen, let us compute!'' By the beginning of this century symbolic logic had progressed to such an extent that the German mathematician David Hilbert declared that all mathematical questions are in principle decidable, and he confidently set out to codify once and for all the methods of mathematical reasoning. This result, which is part of a body of work called algorithmic information theory, is not a cause for pessimism; it does not portend anarchy or lawlessness in mathematics. (Indeed, most mathematicians continue working on problems as before.) What it means is that mathematical laws of a different kind might have to apply in certain situations: statistical laws. In the same way that it is impossible to predict the exact moment at which an individual atom undergoes radioactive decay, mathematics is sometimes powerless to answer particular questions. Nevertheless, physicists can still make reliable predictions about averages over large ensembles of atoms. Mathematicians may in some cases be limited to a similar approach.

46. Science Search > Diophantine Equations
6.00 Votes 555. 3. Developing A General 2nd Degree diophantine equationx^2 + p = 2^n Methods to solve these equations. http//www

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Diophantus Quadraticus

On-line Pell Equation solver by Michael Zuker. detailed information
Rating: [6.00] Votes: [119]
Egyptian Fractions

Lots of information about Egyptian fractions collected by David Eppstein. detailed information Rating: [6.00] Votes: [555] Developing A General 2nd Degree Diophantine Equation x^2 + p = 2^n Methods to solve these equations. detailed information Rating: [5.00] Votes: [1628] Diophantine Geometry in Characteristic p A survey by José Felipe Voloch. detailed information Rating: [5.00] Votes: [754] 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.

47. Hilbert's Tenth Problem. Diophantine Equations. Part 2. By K.Podnieks
4.4. Diophantine Representation of Solutions of Fermat s Equation. Jones, JP Universaldiophantine equation. Journal of Symbolic Logic, 47 (1982), 549571.
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

48. Detalji ZProjekta
Keywords Explicit solutions of diophantine equations, Diophantine mtuples, ellipticcurves of high rank, Thue equation, systems of simultaneous Pellian

49. Diophantine Equation -- From MathWorld
Number Theory diophantine equations A diophantine equation is an algebraic equation in one or more unknownswith integer coefficients, for which integer solutions are sought.
INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index
ABOUT THIS SITE About MathWorld About the Author
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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

50. 3. Diophantine Equations And Logic
3. diophantine equations and logic. I believe logical statements. A statementcan be seen as a ``diophantine equation where we have.
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.

51. Diophantine Equations Information Sites
Top Science Math Number Theory diophantine equations Developing A General 2ndDegree diophantine equation x^2 + p = 2^n Methods to solve these equations. Search
(Not sure of spelling? Use first letters and * such as abc* or abcd* or abcde*) Match:.. All Any
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Search Words: Top Science Math Number Theory : Diophantine Equations

52. Linear Diophantine Equations
Generally this functions is used in connection with operations on latticesbecause a lattice can be seen as a solution of a diophantine equation.
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

53. Open Problem? Diophantine Equations.
Open problem given a diophantine equation that has a solution for every n eg a+b+c =abc ,compute the expected number of steps until a solution is found.
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

54. Open Problem? Diophantine Equations.
Open problem? diophantine equations. diophantine equations. From Marko Riedel mriedel@xxxxxxxxxxxxx ; Date Fri, 12 Mar 2004 145955 +0100.
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.

55. Diophantine Equations Number Theory Math Science
Developing A General 2nd Degree diophantine equation x^2 + p = 2^n Number Theory diophantine equations. English Deutsch Espa±ol ... Diophantine Equations Diophantine Equations : In this seccionre included any equation or set of equations where the unknowns must be integer numbers. It includes Fermat Last Theorem as a special case. Math Number Theory Diophantine Equations: Equal Sums of Like Powers
Fermat's Last Theorem

Solving General Pell Equations

Math Number Theory Diophantine Equations.
John Robertson's treatise on how to solve Diophantine equations of the form x^2 - dy^2 = N.

Math Number Theory Diophantine Equations.
Sets of numbers such that the product of any two is one less than a square. Diophantus found the rational set 1/16 ( 1, 3, 8, 120, ... ) 33/16, 17/4, 105/16; Fermat the integer set 1, 3, 8, 120.
Rational Triangles
Math Number Theory Diophantine Equations. Triangles in the Euclidean plane such that all three sides are rational. With tables of Heronian and Pythagorean triples. Rational and Integral Points on Higher-dimensional Varieties

56. Diophantine Equation
Diophantine linear equation. ALGORITHM. My Diophant algorithm solves almostalways diophantine equation for N =500 and AI =2*10**9. IMPLEMENTATION.
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
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 Definition Meaning Information Explanation
diophantine equation definition, meaning and explanation and moreabout diophantine equation. Free diophantine equation. definition
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Diophantine equation
In mathematics Diophantine equations are equation s 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. Mathematical logic now has shown that it is hopeless to expect a complete theory. 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

58. Diophantine Equation :: Online Encyclopedia :: Information Genius
diophantine equation. Online Encyclopedia Diophantine values. Examplesof diophantine equations are ax + by = 1 See Bézout s identity.
Quantum Physics Pampered Chef Paintball Guns Cell Phone Reviews ... Science Articles Diophantine equation
Online Encyclopedia

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. Examples of Diophantine equations are
  • ax by = 1: See Bézout's identity x n y n z n : For n =2 there are many solutions ( x y z ), the Pythagorean triples. For larger values of n Fermat's Last Theorem states that no positive integer solutions x y z satisfying the above equation exist. x n y Pell's equation ) which is named, mistakenly, after the English mathematician John Pell. It was studied by Fermat
The depth of the study of general Diophantine equations is shown by the characterisation of Diophantine sets as recursively enumerable The field of Diophantine approximation deals with the cases of Diophantine inequalities : variables are still supposed to be integral, but some coefficients may be irrational numbers, and the equality sign is replaced by upper and lower bounds. This content from wikipedia is licensed under the GNU Free Documentation License Power Supplies Hardware Information Law Advice

59. Diophantine Equation From FOLDOC
diophantine equation. mathematics Equations with integer coefficientsto which integer solutions are sought. Because the results equation

60. DIOPHANTINE EQUATION - Meaning And Definition Of The Word
diophantine equation Dictionary Entry and Meaning. Computing Dictionary. DefinitionEquations with integer coefficients to which integer solutions are sought. equation
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DIOPHANTINE EQUATION: Dictionary Entry and Meaning
Computing Dictionary Definition: Equations with integer coefficients to which integer solutions are sought. Because the results are restricted to integers, different algorithms must be used from those which find real solutions. See Also: mathematics HOME ABOUT HYPERDICTIONARY

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