# What is linear Diophantine equation?

A Linear Diophantine equation (LDE) is an equation with 2 or more integer unknowns and the integer unknowns are each to at most degree of 1. Linear Diophantine equation in two variables takes the form of ax+by=c, where x,y∈Z and a, b, c are integer constants. x and y are unknown variables.

## 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.

## What is non linear Diophantine equation?

A non- linear Diophantine equation is every Diophantine equation which is not linear. For instance, the equation \$x^2 + 3y^3 = 35\$ is a non-linear Diophantine equation.

## What is a Diophantine equation in compiler design?

A Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integral solutions are required. An Integral solution is a solution such that all the unknown variables take only integer values. Given three integers a, b, c representing a linear equation of the form : ax + by = c.

## What are Diophantine equations used for?

Diophantine problems have fewer equations than unknowns and involve finding integers that solve simultaneously all equations. As such systems of equations define algebraic curves, algebraic surfaces, or, more generally, algebraic sets, their study is a part of algebraic geometry that is called Diophantine geometry.

## Who invented Diophantine equation?

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.

## How do you find the solution of an integer equation?

Let a,b∈Z with a≠0.
1. If a divides b, then the equation ax=b has exactly one solution that is an integer.
2. If a does not divide b, then the equation ax=b has no solution that is an integer.

## How do you solve linear congruences examples?

The most commonly used methods are the Euclidean Algorithm Method and the Euler's Method.
1. Example: Solve the linear congruence ax = b (mod m)
2. Solution: ax = b (mod m) _____ (1)
3. Example: Solve the linear congruence 3x = 12 (mod 6)
4. Solution:
5. Example: Solve the Linear Congruence 11x = 1 mod 23.

## How do you say Diophantine?

Break 'Diophantine' down into sounds: [DY] + [OH] + [FAN] + [TYN] - say it out loud and exaggerate the sounds until you can consistently produce them. Record yourself saying 'Diophantine' in full sentences, then watch yourself and listen.

## What is x3 y3 z3 K?

The equation x3+y3+z3=k is known as the sum of cubes problem. While seemingly straightforward, the equation becomes exponentially difficult to solve when framed as a "Diophantine equation" -- a problem that stipulates that, for any value of k, the values for x, y, and z must each be whole numbers.

## How can Euclidean algorithm be used to solve linear congruences?

Using Euclid's extended algorithm, we find an integer x0 such that ax0 ≡ 1 (mod m). (1) Such an x0 is called an inverse of a modulo m. 2. Multiplying equation (1) by b, we obtain a(x0b) ≡ b (mod m) so that x = x0b is a solution of the linear congruence.

## How do you solve congruences mod?

To solve a linear congruence ax ≡ b (mod N), you can multiply by the inverse of a if gcd(a,N) = 1; otherwise, more care is needed, and there will either be no solutions or several (exactly gcd(a,N) total) solutions for x mod N.

## Which of the following Diophantine equation is not solvable?

gcd(6, 51) = 3Hence the equation is not solvable. gcd(33, 14) = 1.

## How do you know if a Diophantine equation has infinite solutions?

Let a, b and c be integers with a≠0 and b≠0, and let d=gcd(a,b).
• If d does not divide c, then the linear Diophantine equation ax+by=c has no solution.
• If d divides c, then the linear Diophantine equation ax+by=c has infinitely many solutions.

## Which of the following Diophantine equations has no solution in integers?

Expert Answer. The diophantine equations are of the form a x + b y = c , if c can be divided by the greatest common divisor (gcd) of a and b then this equation has integer solutions. 22 cannot be divided by 3 without forming fractions, so this equation doesn't have any solutions.

## Who is the father of mathematics?

Archimedes is known as the Father Of Mathematics. He lived between 287 BC – 212 BC. Syracuse, the Greek island of Sicily was his birthplace. Archimedes was serving the King Hiero II of Syracuse by solving mathematical problems and by developing interesting innovations for the king and his army.

## Who invented zero?

"Zero and its operation are first defined by [Hindu astronomer and mathematician] Brahmagupta in 628," said Gobets. He developed a symbol for zero: a dot underneath numbers.

## Who is the father of algebra?

al-Khwārizmī, in full Muḥammad ibn Mūsā al-Khwārizmī, (born c. 780 —died c. 850), Muslim mathematician and astronomer whose major works introduced Hindu-Arabic numerals and the concepts of algebra into European mathematics.

## How many solutions does ax by C have?

It has infinitely many solutions.

## How do you solve indeterminate equations?

Indeterminate equations of first degree for special conditions. Solve in positive integers 14x−11y=29. The two equations above are called general solutions. By substituting any positive integral value of p or zero, we get positive integral values of x and y.

## Does 14x 21y C have integer solution?

14x + 21y = 77

The given equation has Infinite solutions.
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