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**individ****Member**- Registered: 2014-03-16
- Posts: 321

For odd numbers.

Decompose the number into factors.

The solution can be written as.

It is seen that for all odd numbers are infinitely many solutions, not just

.*Last edited by individ (2017-03-04 21:21:45)*

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**individ****Member**- Registered: 2014-03-16
- Posts: 321

For the equation.

You can write such a parameterization.

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**individ****Member**- Registered: 2014-03-16
- Posts: 321

For the solution of the equation.

You must use the solutions of the equation.

You can use solutions which are recorded in the subject.

Then using the solutions of this equation can be substituted into the formula and find us.

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**individ****Member**- Registered: 2014-03-16
- Posts: 321

For the solution of the equation.

Solution we write expanding on the multipliers.

And ;Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 321

Write solutions like this.

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**individ****Member**- Registered: 2014-03-16
- Posts: 321

Spread by multipliers...

Decisions can be recorded.

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**NakulG****Member**- Registered: 2014-09-02
- Posts: 186

Hi can you suggest a good charting software, i am trying to plot conic section, intersection of cylindrical cones and a plane. If you can suggest a good charting software that would be great. I tried Asymptote and geogebra... not working out for me.

thanks

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**individ****Member**- Registered: 2014-03-16
- Posts: 321

NakulG wrote:

Hi can you suggest a good charting software, i am trying to plot conic section, intersection of cylindrical cones and a plane. If you can suggest a good charting software that would be great. I tried Asymptote and geogebra... not working out for me.

thanks

Do not write the question here! If you need to open a separate topic. There can be about a specific equation to tell.

I solve the equation. Programs do not write.

Build a curve at the intersection of two surfaces of a standard task. What is the program?

If you need to find the integral points. Formulate problem as a system of nonlinear equations. Then it will try to solve it.

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**individ****Member**- Registered: 2014-03-16
- Posts: 321

For the equation.

We use the solutions of the equation Pell.

Then the solution can write.

And again.

All the numbers can have any sign.

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**individ****Member**- Registered: 2014-03-16
- Posts: 321

Better system to write this.

For simplicity, the formula is written. Solution write using other parameters which will be determined next.

Express these options through the following polynomial.

Express these options through the following polynomial.

And now Express them via the parameters set by us

.The formula is very cumbersome and I wasn't able to test it. We must believe not made any mistake?

More simple parameterizations.

*Last edited by individ (2017-07-02 23:50:04)*

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**individ****Member**- Registered: 2014-03-16
- Posts: 321

There is one idea. To search for the solution of the equation.

If we know any solution

of this equation. Then it is possible to find another . Make such a change.Then the following solution can be found by the formula.

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**individ****Member**- Registered: 2014-03-16
- Posts: 321

Rewrite the system of equations.

The solution of this system can be written as.

*Last edited by individ (2017-08-09 02:04:08)*

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**individ****Member**- Registered: 2014-03-16
- Posts: 321

For the equation.

It is possible to reduce the parameterization of the solutions to some equivalent to the Pell equation.

It has the form.

These parameters can be recorded through the solution of the equation Pell.

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**individ****Member**- Registered: 2014-03-16
- Posts: 321

A better equation to solve in General.

As already mentioned, the task is reduced to some equivalent to the Pell equation. Actually reduced to this form.

Solution we write.

Or so.

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**individ****Member**- Registered: 2014-03-16
- Posts: 321

*Last edited by individ (2017-09-05 19:56:34)*

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**individ****Member**- Registered: 2014-03-16
- Posts: 321

Solution we write.

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**individ****Member**- Registered: 2014-03-16
- Posts: 321

For the equation.

All variations of the same formula. As the number of solutions of course need to consider all possible factorization. Too much ends quickly. The number must be greater than 1.

Consider this example.

;Let

Means ; ;Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 321

For the system of equations.

Make such a change.

The solution can be written as.

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**individ****Member**- Registered: 2014-03-16
- Posts: 321

For the equation.

The solution can be written directly.

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**individ****Member**- Registered: 2014-03-16
- Posts: 321

For this type of equations - which has something in common...

Record regarding

- A different equation. And replace solving a system of equations, the solution of the same equation.We write as....

And instead of solving the system, we solve it.

You can write a simple solution:

- any integer asked us.A similar method can be used for other type of equations.

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**individ****Member**- Registered: 2014-03-16
- Posts: 321

The solution of the system of equations can be written in this form.

are any integer.Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 321

To solve the Diophantine equation.

It is necessary to use the solutions of the Pell equation.

Then the solutions can be written as follows.

For positive you need to take decisions at

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**individ****Member**- Registered: 2014-03-16
- Posts: 321

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**individ****Member**- Registered: 2014-03-16
- Posts: 321

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**individ****Member**- Registered: 2014-03-16
- Posts: 321

You can record such a parameterization of solutions.

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