Solve triple square Diophantine equation

[individ]

Once you know how to solve it, then explain how to solve Diophantine equation:

\(\displaystyle X^2+Y^2=aZ^2\)

\(\displaystyle a\) - integer. Write the equation when it has a solution.

 

[individ]

Thank you!
When you need an answer I will give a link to it.

 

[individ]

For example for such equation:

\(\displaystyle y^2+ax^2=z^2\)

The solutions have the form:

\(\displaystyle y=p^2-as^2\)

\(\displaystyle x=2ps\)

\(\displaystyle z=p^2+as^2\)

For example for such equation:

\(\displaystyle y^2+ax^2=az^2\)

The solutions have the form:

\(\displaystyle y=2aps\)

\(\displaystyle x=ap^2-s^2\)

\(\displaystyle z=ap^2+s^2\)

\(\displaystyle p,s\) - integers.
But for such equations, of which I said, to write a simple formula is impossible. I wonder why?

 

[individ]

You can write and easy solution and similar equation:

number theory - Find all integers satisfying $m^2=n_1^2+n_1n_2+n_2^2$ - Mathematics Stack Exchange

There are many formulas to see. The questions will be answered here.

 

[CellCoree]

The triple square Diophantine equation is a special type of Diophantine equation that involves three variables and squares in the equation. It can be written as:

X^2 + Y^2 = aZ^2

To solve this equation, we first need to understand the properties of the solutions. In order for this equation to have a solution, the value of a must be a perfect square. This means that it can be written as a product of two equal numbers, such as a = n^2, where n is an integer.

Once we have determined that a is a perfect square, we can then proceed to solve the equation. One method to solve this equation is to use the method of infinite descent. This involves finding a solution to the equation and then using that solution to find a smaller solution, repeating the process until we reach the smallest possible solution.

For example, let's say we have the equation X^2 + Y^2 = 25Z^2, where a = 25. We can start by finding a solution to this equation, such as X = 3, Y = 4, and Z = 1. This gives us a solution of 3^2 + 4^2 = 25(1^2), which simplifies to 25 = 25.

Next, we can use this solution to find a smaller solution. We can do this by dividing both sides of the equation by 5, giving us the new equation (3/5)^2 + (4/5)^2 = Z^2. This gives us a new solution of X = 3/5, Y = 4/5, and Z = 1. We can continue this process, finding smaller and smaller solutions until we reach the smallest possible solution, which would be X = 0, Y = 0, and Z = 1.

In general, the method of infinite descent can be used to solve triple square Diophantine equations. Other methods such as the method of Pythagorean triples and the method of Pell's equation can also be used to solve these types of equations.

To solve a general Diophantine equation, we can use a variety of methods depending on the specific equation. Some common methods include substitution, factorization, and modular arithmetic. It is important to first analyze the equation and determine any special properties or restrictions on the solutions before selecting a method to solve it.