What Are the Solutions to the Diophantine Equation x^2 = 2y^2 = 3z^2?

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In summary, the task is to find all solutions to the equation x^2 = 2y^2 = 3z^2. The suggested method is to substitute this formula into the form ap^2 + q^2 = z^2 and solve for x and y in terms of integers. However, this approach does not seem to yield valid solutions.
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lei123
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Homework Statement


x^2 = 2y^2 = 3z^2
Find all the solutions.


Homework Equations


There is a method once I substitute the above formula into the form ap^2 + q^2 = z^2, to get all the solutions.


The Attempt at a Solution


I'm having a little trouble with the substitution.
I was thinking, at first that one could argue that x and y are both congruent to 0, mod 3. Then x-y = 3p and x+ 2y = 3q; one could solve for x, and y, and then backsubstitute, when I get the formulas for x and y in terms of integers, they don't check in the first equation
 
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this problem did not get any feedback?
 
  • #3
lei123 said:

Homework Statement


x^2 = 2y^2 = 3z^2
Are you sure this is correct? Is it supposed to be ##x^2 + 2y^2 =3z^2## or something similar?
##x^2 = 2y^2 = 3z^2## does not seem correct, maybe it's a typo.
 

FAQ: What Are the Solutions to the Diophantine Equation x^2 = 2y^2 = 3z^2?

What is a Diophantine equation?

A Diophantine equation is a polynomial equation in two or more unknowns where the coefficients and the solutions are restricted to integers. These equations are named after the ancient Greek mathematician Diophantus and have been studied for centuries.

How are Diophantine equations different from other types of equations?

Diophantine equations differ from other types of equations in that they only have solutions in integers, while other equations may have solutions in real or complex numbers. They also often involve multiple unknown variables and can have an infinite number of solutions.

What is the significance of Diophantine equations?

Diophantine equations have been studied for their mathematical beauty and complexity, but they also have practical applications in fields such as cryptography and coding theory. They also play a role in number theory and have connections to other areas of mathematics such as geometry and algebra.

Are all Diophantine equations solvable?

No, not all Diophantine equations are solvable. In fact, it has been proven that there is no general algorithm to solve all Diophantine equations. Some equations may have a finite number of solutions, while others may have an infinite number of solutions.

How are Diophantine equations solved?

There are several methods for solving Diophantine equations, depending on the specific equation. Some common techniques include using modular arithmetic, factoring, and applying number theory concepts such as the Euclidean algorithm. In some cases, computer algorithms and programs can also be used to find solutions.

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