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.

]]>For the equation.

If you know any solution

of this equation. Then the formula for the solution of the equation can be written immediately. - any integers.It is seen that such formulas can be written infinitely many. If so like to use the well-known decision, it is better to write like this. This will allow to solve the equation.

It is easy to see that there are solutions for any

. Ask yourself the number . And place into factors.Then the coefficient is set to

us so.Finding all the factors of

you can write a formula for the parameterization of the solution of this equation.The transition to 3 parameters

- Allows not to use too much, but to write formulas.]]>