How do you prove a Diophantine equation?
d=as+bt. dm=(as+bt)mc=a(sm)+b(tm). This means that x=sm, y=tm is a solution of ax+by=c, and we have proved that the Diophantine equation ax+by=c has at least one solution. x=x0+bdk y=y0−adk.
What is quadratic Diophantine equation?
The quadratic diophantine equations are equations of the type: a x 2 + b x y + c y 2 = d where , , and are integers, and we ask the solutions and to be integers.
Who discovered Diophantine equation?
Diophantus of Alexandria
The first known study of Diophantine equations was by its namesake Diophantus of Alexandria, a 3rd century mathematician who also introduced symbolisms into algebra. He was author of a series of books called Arithmetica, many of which are now lost.
What is the use of Diophantine equation?
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 answer to the Diophantine 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.
Why are Diophantine equations important?
The purpose of any Diophantine equation is to solve for all the unknowns in the problem. Indeterminate equations of the second degree or higher contain two or more unknowns to solve for. Diophantine equations are equations of polynomial expressions for which rational or integer solutions are sought.
Why Diophantine equation is important?
What is the use of Diophantine equations?
The mathematical method of diophantine equations is shown to apply to two problems in chemistry: the balancing of chemical equations, and determining the molecular formula of a compound.
What was Diophantus famous for?
Diophantus, byname Diophantus of Alexandria, (flourished c. ce 250), Greek mathematician, famous for his work in algebra.
Why are diophantine equations important?
What is the application of Diophantine equation?
Which of the following is called the 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.
What are real life applications of Diophantine equations?
Are there any solutions?
What was the motivation of studying Diophantine equations?
x = 84 (his age). The motivation to study more equations in one or more unknowns having only integral solutions lead to the origin of Diophantine Equation which is de ned as a Polynomial equation with Integral coe cients which is solvable in Integers. 1.1. Forms (Types) of Diophantine Equations
How to convert a Diophantine equation into parametric form?
– Linear Combination. A Diophantine equation in the form is known as a linear combination. – Pythagorean Triples. A Pythagorean triple is a set of three integers that satisfy the Pythagorean Theorem, . – Sum of Fourth Powers. – Pell Equations. – Methods of Solving. – Fermat’s Last Theorem. – Problems.
Can inequalities be considered in a Diophantine equation?
If a Diophantine equation has as an additional variable or variables occurring as exponents, it is an exponential Diophantine equation. Examples include the Ramanujan–Nagell equation , 2 n − 7 = x 2 , and the equation of the Fermat–Catalan conjecture and Beal’s conjecture , a m + b n = c k with inequality restrictions on the exponents.