How do you know if a Diophantine equation has no solution?

How do you know if a Diophantine equation has no solution?

Applied to the simplest Diophantine equation, ax + by = c, where a, b, and c are nonzero integers, these methods show that the equation has either no solutions or infinitely many, according to whether the greatest common divisor (GCD) of a and b divides c: if not, there are no solutions; if it does, there are …

Is Diophantine equation solvable?

For instance, we know that linear Diophantine equations are solvable.

What is Diophantine equation used for?

In mathematics diophantine equations are central objects in number theory as they express natural questions such as the ways to write a number as a sum of cubes, but they naturally come up in all questions that can be reduced to questions involving discrete objects, e.g. in algebraic topology.

What is the purpose of Diophantine equation?

The purpose of any Diophantine equation is to solve for all the unknowns in the problem. When Diophantus was dealing with 2 or more unknowns, he would try to write all the unknowns in terms of only one of them.

How do you find the solution of an integral of an equation?

Equation type: Ax + By = C

1. First, reduce the equation in lowest reducible form.
2. After reducing, if coefficients of x and y still have a common factor, the equation will have no solutions.
3. If x and y are co-prime in the lowest reducible form, find any one integral solution.

What is meant by Diophantine equation?

In mathematics, a Diophantine equation is a polynomial equation, usually involving two or more unknowns, such that the only solutions of interest are the integer ones (an integer solution is such that all the unknowns take integer values).

What are Diophantine equations used for?

How many variables are there in linear Diophantine equation?

two variables
Definition: linear Diophantine equation in two variables Let a, b, and c be integers with a≠0 and b≠0. The Diophantine equation ax+by=c is called a linear Diophantine equation in two variables.

What is Diophantine analysis?

or Diophantine analysis noun Mathematics. any of several methods for finding integral solutions for equations with more than one variable whose coefficients are integers.

Is there a general solution to a Diophantine equation?

Techniques such as infinite Descent can also show that no solutions to a particular equation exist, or that no solutions outside of a particular family exist. It is natural to ask whether there is a general solution for Diophantine equations, i.e., an algorithm that will find the solutions for any given Diophantine equations.

When did Florentin Smarandache write the Diophantine equation?

The book offers solutions to a multitude of \–Diophantine equation proposed by Florentin Smarandache in previous works [Smaran- dache, 1993, 1999b, 2006] over the past two decades.

Is the study of integer solutions called Diophantine analysis?

The study of problems that require integer solutions is often referred to as Diophantine analysis. Although the practical applications of Diophantine analysis have been somewhat limited in the past, this kind of analysis has become much more important in the digital age.