For Diophantine equation

If the root is integer.

We use the solutions of the equation Pell.

The solution then can be written in this form.

To find all solutions is necessary to solve the more General equation. With different coefficients.

]]>You can record a similar system.

Parametrization of solutions we write this.

Consider a special case.

Using the solutions of the equation Pell.

Enough to know first, everything else will find a formula.

The solution then write.

These solutions are negative.

And a positive decision of the same are determined by the Pell equation.

Use the first solution.

Next find the formula.

Will make a replacement.

The decision record.

]]>For such equations.

Solution write.

]]>It turned out the following. For a square shape

If any number of options

and any coefficients . Solutions is always there.Illustrate 2 coefficients. Similarly solved if the number of different factors. That was evident symmetry and do not get confused is better to take 3 equation.

The solution is easy to write.

Here the representation of 3 options, but it is easy to see that can be written in the form of a combination with any number of options.

]]>For the system of equations.

Solutions can be parameterized.

It is interesting that such triples can be too much. The formula can be increased to any number. That is the same to write not only for 4 partitions, but for any number. The main thing that all the variables were not identical to each other.

]]>Solutions have the form.

]]>Solution.

]]>Using the solutions of the equation Pell.

Solution write.

]]>For such systems of equations.

It is better to use an algebraic approach. He gives at once rational decisions.

]]>For Diophantine equation.

You can record a parameterization.

]]>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.

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