Diophantine set

Diophantine equations

In mathematics, a Diophantine equation is an equation of the form P(x1, ..., xj, y1, ..., yk) = 0 (usually abbreviated P(x, y) = 0) where P(x, y) is a polynomial with integer coefficients, where x1, ..., xj indicate parameters and y1, ..., yk indicate unknowns. A Diophantine set is a subset S of , the set of all j-tuples of natural numbers, so that for some Diophantine equation P(x, y) = 0, That is, a parameter value is in the Diophantine set S if and only if the associated Diophantine equation is satisfiable under that parameter value. The use of natural numbers both in S and the existential quantification merely reflects the usual applications in computability and model theory. It does not matter whether natural numbers refer to the set of nonnegative integers or positive integers since the two definitions for Diophantine set are equivalent. We can also equally well speak of Diophantine sets of integers and freely replace quantification over natural numbers with quantification over the integers. Also it is sufficient to assume P is a polynomial over and multiply P by the appropriate denominators to yield integer coefficients. However, whether quantification over rationals can also be substituted for quantification over the integers is a notoriously hard open problem. The MRDP theorem (so named for the initials of the four principal contributors to its solution) states that a set of integers is Diophantine if and only if it is computably enumerable. A set of integers S is computably enumerable if and only if there is an algorithm that, when given an integer, halts if that integer is a member of S and runs forever otherwise. This means that the concept of general Diophantine set, apparently belonging to number theory, can be taken rather in logical or recursion-theoretic terms. This is far from obvious, however, and represented the culmination of some decades of work. Matiyasevich's completion of the MRDP theorem settled Hilbert's tenth problem. Hilbert's tenth problem was to find a general algorithm which can decide whether a given Diophantine equation has a solution among the integers. While Hilbert's tenth problem is not a formal mathematical statement as such, the nearly universal acceptance of the (philosophical) identification of a decision algorithm with a total computable predicate allows us to use the MRDP theorem to conclude that the tenth problem is unsolvable. (Wikipedia).

Dangers of Dihydrogen Monoxide - The Less You Know 523

To learn how you can help in the fight against Dihydrogen Monoxide check out http://www.stevespanglerscience.com/dhmo.html

From playlist April Fool's Day Favorites

Dangers of Dihydrogen Monoxide - The Less You Know 632

To learn how you can help in the fight against Dihydrogen Monoxide check out http://www.stevespanglerscience.com/dhmo.html

From playlist April Fool's Day Favorites

Dangers of Dihydrogen Monoxide - The Less You Know 141

To learn how you can help in the fight against Dihydrogen Monoxide check out http://www.stevespanglerscience.com/dhmo.html

From playlist April Fool's Day Favorites

How to test high voltage diodes

This is a tutorial on how to test if a high voltage diode is good and how to determine the direciton of a high voltage diode, in case it's not marked. High voltage diodes can be bought retail but are also often found in microwave ovens. In the hobbiest world, high voltage diodes are often

From playlist Basic Electronics

Dangers of Dihydrogen Monoxide - The Less You Know 297

To learn how you can help in the fight against Dihydrogen Monoxide check out http://www.stevespanglerscience.com/dhmo.html

From playlist April Fool's Day Favorites

Dangers of Dihydrogen Monoxide - The Less You Know 119

To learn how you can help in the fight against Dihydrogen Monoxide check out http://www.stevespanglerscience.com/dhmo.html

From playlist April Fool's Day Favorites

World's Most Powerful Visible Diode Laser

"The NUBM44 Laser Diode" The World's Most Powerful

From playlist Lasers

Dangers of Dihydrogen Monoxide - The Less You Know 217

To learn how you can help in the fight against Dihydrogen Monoxide check out http://www.stevespanglerscience.com/dhmo.html

From playlist April Fool's Day Favorites

Frank Calegari: 30 years of modularity: number theory since the proof of Fermat's Last Theorem

From playlist IMU Live Events

Diophantine Equations: Polynomials With 1 Unknown ← number theory ← axioms

Learn how to solve a Diophantine Equation that's a polynomial with one variable. We'll cover the algorithm you can use to find any & all integer solutions to these types of equations. written, presented, & produced by Michael Harrison #math #maths #mathematics you can support axioms on

From playlist Number Theory

Diophantine Equation: ax+by=gcd(a,b) ← Number Theory

Once you know how to solve diophantine equations with a single variable, the next step in complexity is to consider equations with two variables. The simplest such equations are linear and take the form ax+by=c. Before we solve this equation generally, we need a preliminary result. We s

From playlist Number Theory

Reducibility for the Quasi-Periodic Liner Schrodinger and Wave Equations - Lars Hakan Eliasson

Lars Hakan Eliasson University of Paris VI; Institute for Advanced Study February 21, 2012 We shall discuss reducibility of these equations on the torus with a small potential that depends quasi-periodically on time. Reducibility amounts to "reduce” the equation to a time-independent linea

From playlist Mathematics

Stars-and-Bars to Find the Number of Solutions of a Diophantine Equation!

In this video, we will be applying a well-known Combinatorics concept, Stars and Bars, to find the number of integer solutions to a Diophantine Equation. We will cover 2 problems: w+x+y+z = 10 Problem 1: how many solutions for w, x, y, z being non-negative integers? Problem 2: how many

From playlist Elementary Number Theory

Hakan Eliasson: Quasi-periodic wave equation - almost reducibility-

Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b

From playlist Dynamical Systems and Ordinary Differential Equations

Dihydrogen Monoxide - Lethal Everyday Chemical

Dihydrogen Monoxide is a chemical that is available to small children without special permission. It can be lethal in small doses, and causes millions of dollars in damage to the environment and personal property each year. Be careful when dealing with this dangerous chemical.

From playlist April Fool's Day Favorites

A Short Course in Algebra and Number Theory - Elementary Number Theory

To supplement a course taught at The University of Queensland's School of Mathematics and Physics I present a very brief summary of algebra and number theory for those students who need to quickly refresh that material or fill in some gaps in their understanding. This is the fourth lectu

From playlist A Short Course in Algebra and Number Theory

Application of a Linear Diophantine Equation: Number of Stamps

This video provides an application of a Diophantine equation that is solved using congruence. mathispower4u.com

From playlist Additional Topics: Generating Functions and Intro to Number Theory (Discrete Math)

Approximating Irrational Numbers (Duffin-Schaeffer Conjecture) - Numberphile

James Maynard recently co-authored a proof of the Duffin-Schaeffer Conjecture. More links & stuff in full description below ↓↓↓ More James Maynard on Numberphile: http://bit.ly/JamesMaynard On the Duffin-Schaeffer conjecture - by Dimitris Koukoulopoulos and James Maynard - https://arxiv.

From playlist James Maynard on Numberphile

Dangers of Dihydrogen Monoxide - The Less You Know 471

To learn how you can help in the fight against Dihydrogen Monoxide check out http://www.stevespanglerscience.com/dhmo.html

From playlist April Fool's Day Favorites

Diophantine approximation and Diophantine definitions - Héctor Pastén Vásquez

Short Talks by Postdoctoral Members Héctor Pastén Vásquez - September 29, 2015 http://www.math.ias.edu/calendar/event/88264/1443549600/1443550500 More videos on http://video.ias.edu

From playlist Short Talks by Postdoctoral Members

Diophantine set

Related pages