For finding solutions in the case if the square is rational.

Use the same formula. For finding all solutions need to consider all possible solutions of the equation Pell. Consider the case where these solutions exist. To do this, lay on the multiplier coefficients.

And you need to check when this Pell equation has the solutions?

And then the solution is substituted in the above formula.

Consider the case which invited.

; ; ; ; ; ;Use the first solution.

;Then ;

]]>For the system of Diophantine equations.

You can write the parameterization of the solutions.

This means that for every Pythagorean triple has infinitely many solutions.

]]>For the system of equations .

The solution can be written in this form.

]]>For the equation.

You can write for example the following parameterization.

turn out negative.Will make a replacement. We introduce the number.

We will use the solutions of Pell's equation.

Knowing the first solution i

, you can find the rest on the previous formula.Now knowing this, you can write down the solutions themselves.

can have any sign.]]>http://math.stackexchange.com/questions … n-integers

For the equation.

You can write the solution in this form.

Such a record is better because allows to solve the symmetrical equation with any number of summands. It is only necessary to increase the number of parameters.

]]>http://math.stackexchange.com/questions … ls#1780576

This is Pell's equation and the task is to reduce it to some kind of equivalent.

If you can imagine

Using solutions of Pell's equation.

Decisions can be recorded.

]]>http://www.artofproblemsolving.com/comm … ber_theory

Solution always can be written, for example.

]]>To describe the solutions of the equation.

I think best would be to describe a solution using

parameters.]]>But apparently the time has not come for them. Even attempt a discussion and a proposal to solve the equations in General form leads to the opposition.

http://mathoverflow.net/questions/23414 … -equations

Attempt to offer to solve an equation of the form

Only causes aggression and a desire to remove. With this notation explicitly perceived very well.

It is seen that the time for the ideas of formalization and a common approach is not yet.

Although attempts to reduce such ideas and methods you need to do.]]>

Write for the equation.

One simple formula.

If you use the solutions of the equation Pell. Where

ask yourself.Make the change.

Then decisions can be recorded.

]]>I think it is better to solve a more formal task. We write the system.

We need to find solutions

- that was an arithmetic progression. This will help the solution of the equation Pell.Knowing any solution of the equation Pell

you can always find the following formula.Having any decision - can immediately write down the solution of this system.

Interesting here is that the

looks like an arithmetic progression.]]>It is better to use a more General approach. We write the system.

If the number

, lay at the multipliers. We find then the desired settings.Then the solution can be written as.

]]>For the equation.

If we use the solutions of Pell equations.

Using my replacement.

Decisions will be.

]]>You can write a fairly simple formula.

]]>A more compact solution.

]]>