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## #401 2017-03-04 21:20:53

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

### Re: Formulas for the solution of Diophantine equations.

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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## #402 2017-03-15 19:35:45

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

### Re: Formulas for the solution of Diophantine equations.

For the equation.

You can write such a parameterization.

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## #403 2017-04-02 00:52:55

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

### Re: Formulas for the solution of Diophantine equations.

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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## #404 2017-04-16 20:06:27

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

### Re: Formulas for the solution of Diophantine equations.

For the solution of the equation.

Solution we write expanding on the multipliers.

And
;

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## #405 2017-04-20 05:14:13

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

### Re: Formulas for the solution of Diophantine equations.

Write solutions like this.

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## #406 2017-05-14 05:41:16

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

### Re: Formulas for the solution of Diophantine equations.

Decisions can be recorded.

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## #407 2017-05-20 01:53:43

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

### Re: Formulas for the solution of Diophantine equations.

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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## #408 2017-05-20 02:08:47

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

### Re: Formulas for the solution of Diophantine equations.

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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## #409 2017-05-26 20:52:06

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

### Re: Formulas for the solution of Diophantine equations.

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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## #410 2017-07-02 01:08:40

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

### Re: Formulas for the solution of Diophantine equations.

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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## #411 2017-07-10 21:18:55

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

### Re: Formulas for the solution of Diophantine equations.

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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## #412 2017-08-09 00:35:43

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

### Re: Formulas for the solution of Diophantine equations.

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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## #413 2017-08-21 03:57:55

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

### Re: Formulas for the solution of Diophantine equations.

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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## #414 2017-08-22 23:40:56

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

### Re: Formulas for the solution of Diophantine equations.

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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## #415 2017-09-05 19:55:20

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

### Re: Formulas for the solution of Diophantine equations.

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

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## #416 2017-09-14 16:44:57

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

### Re: Formulas for the solution of Diophantine equations.

Solution we write.

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## #417 2017-09-18 20:59:29

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

### Re: Formulas for the solution of Diophantine equations.

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
;

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## #418 2017-10-02 21:25:36

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

### Re: Formulas for the solution of Diophantine equations.

For the system of equations.

Make such a change.

The solution can be written as.

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## #419 2017-11-05 22:14:52

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

### Re: Formulas for the solution of Diophantine equations.

For the equation.

The solution can be written directly.

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## #420 2017-12-15 17:14:01

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

### Re: Formulas for the solution of Diophantine equations.

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:

A similar method can be used for other type of equations.

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## #421 2018-01-07 17:35:17

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

### Re: Formulas for the solution of Diophantine equations.

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

are any integer.

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## #422 2018-04-30 20:39:03

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

### Re: Formulas for the solution of Diophantine equations.

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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## #423 2018-06-08 04:49:25

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

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## #424 2018-07-17 02:30:02

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

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## #425 2018-07-26 00:27:30

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

### Re: Formulas for the solution of Diophantine equations.

You can record such a parameterization of solutions.

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