Bézout's identity and Diophantine Equation

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So, any solution to ax+by=d leads to an infinite number of solutions to ax+by+cz=d by adding any multiple of z to x and y.In summary, Bézout's Lemma states that the greatest common divisor of a set of numbers can always be written as a linear combination of those numbers. This applies to all of the equations mentioned in the conversation, except for the equation 2x+4y+6z = 7.
  • #1
haki
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I am having problems with one exam question.

Does this diophantine equation have a solution(s)

12a+21b+33c=6

as far as I know this is not a linear equation, and what I read online says that Bezout identity only applies for linear diophantine equations.

The solution says gcd(12,21,33) = 3, 6|3 the above equation has infinitely many solutions. Is that correct? Am I correct to assume then that

12a+21b+33c+24d = 6 again has infinite solutions aswell?

How about this equation

17x+6y +3z =73

since gcd(17,6,3) = 1 and 73 | 1 this has also infinitely many solutions?
 
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  • #2
Bézout's Lemma says, that the greatest common divisor ##d## of numbers ##a_1,\ldots,a_n## can always be written as ##d= s_1a_1+\ldots +s_na_n##. This immediately applies to all of your examples.

An example where it does not work is ##2x+4y+6z = 7##.
 
  • #3
You can always reduce the case ax+by+cz=d to ax+by=d by letting z=0.
 

1. What is Bézout's identity?

Bézout's identity is a theorem in number theory that states that for any two nonzero integers a and b, there exist integers x and y such that ax + by is the greatest common divisor of a and b.

2. What is a Diophantine equation?

A Diophantine equation is an algebraic equation where the solutions are required to be integers. These types of equations are named after the ancient Greek mathematician Diophantus who first studied them.

3. How is Bézout's identity related to Diophantine equations?

Bézout's identity is often used to find integer solutions to Diophantine equations. By finding the x and y values in ax + by = gcd(a,b), we can find solutions to the Diophantine equation ax + by = c by multiplying both sides by c/gcd(a,b).

4. Can Bézout's identity be extended to more than two integers?

Yes, Bézout's identity can be extended to any number of integers. This is known as the generalized Bézout's identity and states that for any set of integers a1, a2, ..., an, there exist integers x1, x2, ..., xn such that a1x1 + a2x2 + ... + anxn is the greatest common divisor of a1, a2, ..., an.

5. What are some real-world applications of Bézout's identity and Diophantine equations?

Bézout's identity and Diophantine equations have many applications in fields such as cryptography, coding theory, and number theory. They can also be used to solve problems in engineering and physics, such as finding integer solutions to equations representing electrical circuits or physical systems.

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