How do you find the general solution of a Diophantine equation?

For example,

Input: 25x + 10y = 15.
Output: General Solution of the given equation is. x = 3 + 2k for any integer m. y = -6 – 5k for any integer m.
Input: 21x + 14y = 35.
Output: General Solution of the given equation is. x = 5 + 2k for any integer m. y = -5 – 3k for any integer m.

Is Diophantine equation solvable?

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

How many solutions are there to the Diophantine equation?

In the example above, an initial solution was found to a linear Diophantine equation. This is just one solution of the equation, however. When integer solutions exist to an equation a x + b y = n , ax+by=n, ax+by=n, there exist infinitely many solutions.

What is meant by Diophantine equation?

Diophantine equation, equation involving only sums, products, and powers in which all the constants are integers and the only solutions of interest are integers. For example, 3x + 7y = 1 or x2 − y2 = z3, where x, y, and z are integers.

How do you calculate Diophantine?

One equation The simplest linear Diophantine equation takes the form ax + by = c, where a, b and c are given integers. The solutions are described by the following theorem: This Diophantine equation has a solution (where x and y are integers) if and only if c is a multiple of the greatest common divisor of a and b.

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.

What is Diophantine equation with example?

What are diophantine equations used for?

Why we use linear Diophantine equation?

For linear Diophantine equation equations, integral solutions exist if and only if, the GCD of coefficients of the two variables divides the constant term perfectly. In other words the integral solution exists if, GCD(a ,b) divides c.