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

Developing A General 2 nd Degree Diophantine Equation x + p = 2 n Thien Do Westmoore High School Science Department Oklahoma City, Oklahoma 73170 Abstract Introduction Methodology Conclusions ... References Abstract 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. Introduction 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 Category Science Math Number Theory diophantine equations......Solves quadratic diophantine equations (integer equations of the form a x^2 + b xy + c y^2 + dx + ey + f = 0) http://www.alpertron.com.ar/QUAD.HTM

If you are using that software, you should enable JavaScript, and then reload this page.

3. Diophantine Equation -- From MathWorld An equation in which only integer solutions are allowed. Hilbert's 10th problem asked if a technique for solving a general Diophantine existed. A general method exists for the solution of first degree diophantine equations. http://mathworld.wolfram.com/DiophantineEquation.html

Number Theory Diophantine Equations Diophantine Equation An equation in which only integer solutions are allowed. Hilbert's 10th problem asked if a technique for solving a general Diophantine existed. A general method exists for the solution of first degree Diophantine equations. However, the impossibility of obtaining a general solution was proven by Julia Robinson and Martin Davis in 1970, following proof of the result that the relation (where is a Fibonacci number ) is Diophantine by Yuri Matiyasevich (Matiyasevich 1970, Davis 1973, Davis and Hersh 1973, Davis 1982, Matiyasevich 1993). 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 iff there exist integers x y z , ... such that Jones and Matiyasevich (1982) proved that no algorithms can exist to determine if an arbitrary Diophantine equation in nine variables has solutions. As a consequence of this result, it can be proved that there does not exists a general algorithm for solving a quartic Diophantine equation , although the algorithm for constructing such an unsolvable quartic Diophantine equation can require arbitrarily many variables (Matiyasevich 1993).

4. Diophantine Equation. The American Heritage® Dictionary Of The English Language diophantine equation. NOUN An algebraic equation with two or more variables whose coefficients are integers, studied http://www.bartleby.com/61/55/D0235550.html

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5. Diophantine Equation diophantine equation. Introduction. A diophantine equation is an equation in which only Integer solutions are allowed. http://granite.sru.edu/~venkatra/Diophantine_Equation.htm

Diophantine Equation Introduction A Diophantine equation is an equation in which only Integer solutions are allowed. Hilbert's 10th Problem asked if a technique for solving a general Diophantine existed. A set of (originally) unsolved problems in mathematics proposed by Hilbert. Of the 23 total, ten were presented at the Second International Congress in Paris on August 8 1900. These problems were designed to serve as examples for the kinds of problems whose solutions would lead to the furthering of disciplines in mathematics. A professor at the University of Gottingen, Hilbert was an extraordinary generalist with a passion for order and rigor. He was just the man to wake other mathematicians to take notice. Some problems Hilbert proposed to the congress (such as number four, the "problem of the straight line as the shortest distance between two points") reflected his own back-to-basics approach to mathematics. Problem ten dealt with Diophantine equations, algebraic equations in several variables whose solutions are required to be rational numbers – that is, whole numbers or fractions, the ratio of whole numbers. Diophantine equations take their name from the Greek mathematician Diophantus of Alexandria, who probably lived in the third century of our era and who discussed such problems at length in his treatise Arithmetica. Hilbert’s tenth problem posed a challenge of breathtaking generality with Diophantus. Some time around 1637, Pierre de Fermat, a French provincial lawyer and passionate amateur mathematician, encountered Diophantus’s result in his copy of a translation of Diophantus. This is when the seed planted by Diophantus came into bloom.

6. Mathematical Applets Draw a graph y = f(x) or fractal, seek prime numbers or solve the diophantine equation. http://www.kolumbus.fi/hoijer.heikki/

7. Randomness In Arithmetic The mathematical assertion that the diophantine equation with parameter k has no solution encodes the assertion that the http://www.umcs.maine.edu/~chaitin/sciamer2.html

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.

8. Diophantine Equation--4th Powers -- From MathWorld diophantine equation4th Powers, As a consequence of Matiyasevich'srefutation of Hilbert's 10th problem, it can be proved that http://mathworld.wolfram.com/DiophantineEquation4thPowers.html

Number Theory Diophantine Equations Diophantine Equation4th Powers As a consequence of Matiyasevich's refutation of Hilbert's 10th problem, it can be proved that there does not exists a general algorithm for solving a general quartic Diophantine equation. However, the algorithm for constructing such an unsolvable quartic Diophantine equation can require arbitrarily many variables (Matiyasevich 1993). As a part of the study of Waring's problem , it is known that every positive integer is a sum of no more than 19 positive biquadrates ( ), that every "sufficiently large" integer is a sum of no more than 16 positive biquadrates ( ), and that every integer is a sum of at most 10 signed biquadrates ( ; although it is not known if 10 can be reduced to 9). The first few numbers n which are a sum of four fourth powers equations) are 353, 651, 2487, 2501, 2829, ... (Sloane's The 4.1.2 equation is a case of Fermat's last theorem with n = 4 and therefore has no solutions. In fact, the equations also have no solutions in integers (Nagell 1951, pp. 227 and 229). The equation

9. LINEAR DIOPHANTINE EQUATIONS A web tool for solving diophantine equations of the form ax + by = c. http://thoralf2.uwaterloo.ca/htdocs/linear.html

Solving ax +by = c a b c

10. Diophantine Equation From FOLDOC diophantine equation. mathematics Equations with integer coefficientsto which integer solutions are sought. Because the results http://wombat.doc.ic.ac.uk/foldoc/foldoc.cgi?Diophantine equation

11. Hilbert's Tenth Problem. Diophantine Equations 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

Hilbert, tenth problem, 10th, problem, Diophantine, equation Back to title page Left Adjust your browser window Right 4. Hilbert's Tenth Problem 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 See also a collection of Hilbert web-sites at www.geometry.net 4.1. History of the Problem. Story of the Solution 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:

12. Generation5.org - Genetic Algorithm Example: Diophantine Equation Genetic Algorithm Example diophantine equation. Make sure that you haveread the genetic algorithms essay before reading this example. http://www.generation5.org/gaexample.shtml

Home Essays Interviews Programs ... Search Search: Genetic Algorithm Example: Diophantine Equation Make sure that you have read the genetic algorithms essay before reading this example. You also must have a working knowledge of C++ and object-oriented programming to utilize the classes and code examples provided. Genetic Algorithm Example Let us consider a diophantine (only integer solutions) equation: a+2b+3c+4d=30 , where a,b,c,d are positive integers. Using a genetic algorithm, all that is needed is a little time to reach a solution (a,b,c,d) . Of course you could ask, why not just use a brute force method (plug in every possible value for a,b,c,d given the constraints < a,b,c,d = )? The architecture of GA systems allow for a solution to be reached quicker since "better" solutions have a better chance of surviving and procreating, as opposed to randomly throwing out solutions and seeing which ones work. Let's start from the beginning. First we will choose 5 random initial solution sets, with constraints < a,b,c,d =

13. 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

Solving the generalized Pell equation x - Dy =N John Robertson An improved version of what used to be here is now a PDF file at Solving the generalized Pell equation. The old HTML page (uncorrected, unenhanced, on some browsers some math symbols do not display correctly) is at old HTML page. This page is best viewed using Microsoft Internet Explorer (MS IE). Last Modified July 3, 2002 John P. Robertson JPR2718@AOL.COM This page has been visited times.

14. The Prime Glossary: Diophantus Now we call an equation to be solved in integers a diophantine equation. For example, Diophantus considered the http://www.utm.edu/research/primes/glossary/Diophantus.html

Diophantus (another Prime Pages ' Glossary entries) Glossary: Index Search Home Previous ... Mail Editor Prime Pages: Home Search Index Diophantus has come to be called the "Father of Geometry." He lived during the period from 250 to 350 A.D., "a silver age in mathematics." His text the Arithmetica was composed of 13 books and 189 problems. The problems he worked on were mostly linear systems of equations with a few quadratics. He included strong hints to make the questions easier to solve. One of these problems uses his age as a solution, so he apparantly lived to at least 84. Diophantus introduced symbols for subtraction, for an unknown, and for the degree of the variable. Although there were several solutions to some of his problems, he only looked for one positive integer solution. Now we call an equation to be solved in integers a diophantine equation . For example, Diophantus considered the equations ax by c where the variables x and y are positive integers. This equation is solvable if and only if the greatest common divisor of a and b divides c . We can find the solution to these equations using a modified Euclidean Algorithm This entry edited from a contribution by Jimmy Goodman.

15. Generation5.org - Essays Genetic Algorithms diophantine equation Solver. This is a C++ programthat solves a diophantine equation using genetic algorithms. http://www.generation5.org/diophantine_ga.shtml

Home Essays Interviews Programs ... Search Search: Genetic Algorithms: Diophantine Equation Solver This is a C++ program that solves a diophantine equation using genetic algorithms. This is the first program is a new set of programs popping up on Generation5 - case studies. Therefore this page is a huge break down of the code and what it does, how it does it, and how to use it. Without further ado, here is the download link: diophantine.zip This code is the accompanying code to the genetic algorithms example , so please read that if you don't know anything about GAs before attempting to look at this code. Trust me, Sam wrote the essay, and I learnt and coded the program straight from what I learnt from that essay - its great! Now for the code: CDiophantine Firstly the class header (note for formatting reasons, a lot of the documentation is taken out): Firstly you notice that there are two structures, the gene structure and the actual CDiophantine class. The gene structure is used to keep track of the different solution sets. The population generated is a population of genes. The gene structure keeps track of its own fitness and likelihood values itself. I also coded a small function to test for equality, this just made some other code a lot more concise. Now onto the functions. Fitness function The fitness functions calculate the fitness of each gene. In our case the fitness function is the difference between the calculated value of the gene and the result we want. This class uses two functions, one that calculates all the fitnesses and another smaller one (you should probably make the function inline) to calculate it per gene.

16. Mathsoft: Mathsoft Unsolved Problems: On A Generalized Fermat-Wiles Equation Steven Finch's essay on the diophantine equation of the form x^n + y^n = c.z^n. http://www.mathsoft.com/mathresources/problems/article/0,,2186,00.html

search site map about us  + news  + ... Unsolved Problems Links On a Generalized Fermat-Wiles Equation Zero Divisor Structure in Real Algebras Sleeping Habits of Armadillos Engineering Standards Mathsoft Constants ... Math Resources On a Generalized Fermat-Wiles Equation Fermat's Last Theorem was no more than a conjecture for over 350 years. Let n be an integer greater than 2. Fermat claimed that any integers x, y and z, not necessarily positive, for which x n + y n = z n To some people, the passage of this conjecture to theoremhood is marked by sadness. They may mistakenly believe that no other interesting Diophantine equations are left to be solved. This essay is aimed at such individuals: there is a much larger class of equations, of which Fermat-Wiles is only a special case, that is well worth everyone's attention! The equation we'll examine is x n + y n n where c is a positive integer. We wish to learn what conditions on n and c force the existence of a non-trivial 0. In other words, when is the equation x n + y n n solvable x + y Click here for more terms The infinite sequence of all such integers c can be shown to be highly sparse relative to the sequence of positive integers. By the

17. Diophantine Equation - Wikipedia diophantine equation. From Wikipedia, the free encyclopedia. Diophantineequations are equations of the form f = 0, where f is a http://www.wikipedia.org/wiki/Diophantine_equation

18. The Beal Conjecture $75,000 prized problem pertaining to the diophantine equation of the form A^x + B^y = C^z where A, B, C, x, y and z are positive integers and x, y and z are all greater than 2, then A, B and C must have a common factor. http://www.math.unt.edu/~mauldin/beal.html

THE BEAL CONJECTURE AND PRIZE BEAL'S CONJECTURE: If A x +B y = C z , where A, B, C, x, y and z are positive integers and x, y and z are all greater than 2, then A, B and C must have a common prime factor. THE BEAL PRIZE. The conjecture and prize was announced in the December 1997 issue of the Notices of the American Mathematical Society. Since that time Andy Beal has increased the amount of the prize for his conjecture. The prize is now this: $100,000 for either a proof or a counterexample of his conjecture. The prize money is being held by the American Mathematical Society until it is awarded. In the meantime the interest is being used to fund some AMS activities and the annual Erdos Memorial Lecture. CONDITIONS FOR WINNING THE PRIZE. The prize will be awarded by the prize committee appointed by the American Mathematical Society. The present committee members are Charles Fefferman, Ron Graham, and Dan Mauldin. The requirements for the award are that in the judgment of the committee, the solution has been recognized by the mathematics community. This includes that either a proof has been given and the result has appeared in a reputable refereed journal or a counterexample has been given and verified. PRELIMINARY RESULTS.

19. 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. http://www.cs.auckland.ac.nz/CDMTCS/chaitin/sciamer2.html

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.

20. Diophantine Equation (FWD) 27 Nov 1998 diophantine equation (FWD), by Antreas P. Hatzipolakis http://mathforum.com/epigone/math-history-list/gandplyston

a topic from math-history-list diophantine equation (FWD) post a message on this topic post a message on a new topic 27 Nov 1998 diophantine equation (FWD) , by Antreas P. Hatzipolakis 27 Nov 1998 Re: diophantine equation (FWD) , by John Conway 27 Nov 1998 Re: diophantine equation (FWD) , by Bill Dubuque 28 Nov 1998 Re: diophantine equation (FWD) , by Antreas P. Hatzipolakis 28 Nov 1998 Re: diophantine equation (FWD) , by Antreas P. Hatzipolakis 28 Nov 1998 Re: diophantine equation (FWD) , by John Conway The Math Forum

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