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individ

Member

Registered: 2014-03-16

Posts: 339

Re: Formulas for the solution of Diophantine equations.

http://math.stackexchange.com/questions … rfect-squa

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)

individ

Member

Registered: 2014-03-16

Posts: 339

Re: Formulas for the solution of Diophantine equations.

http://math.stackexchange.com/questions … te-integer

For the equation.

You can write such a parameterization.

individ

Member

Registered: 2014-03-16

Posts: 339

Re: Formulas for the solution of Diophantine equations.

http://math.stackexchange.com/questions … 24#2214424

For the solution of the equation. 

You must use the solutions of the equation. 

You can use solutions which are recorded in the subject.  http://math.stackexchange.com/questions … 219#709219

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

individ

Member

Registered: 2014-03-16

Posts: 339

Re: Formulas for the solution of Diophantine equations.

https://math.stackexchange.com/question … e-equation

For the solution of the equation.

Solution we write expanding on the multipliers.

And

;

individ

Member

Registered: 2014-03-16

Posts: 339

Re: Formulas for the solution of Diophantine equations.

https://math.stackexchange.com/question … x2y2-z2w21

Write solutions like this.

individ

Member

Registered: 2014-03-16

Posts: 339

Re: Formulas for the solution of Diophantine equations.

https://math.stackexchange.com/question … 01#2280801

Spread by multipliers... 

Decisions can be recorded.

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

individ

Member

Registered: 2014-03-16

Posts: 339

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

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.

individ

Member

Registered: 2014-03-16

Posts: 339

Re: Formulas for the solution of Diophantine equations.

https://math.stackexchange.com/question … m3n3-holds

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.

individ

Member

Registered: 2014-03-16

Posts: 339

Re: Formulas for the solution of Diophantine equations.

https://math.stackexchange.com/question … 18#2343918

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)

individ

Member

Registered: 2014-03-16

Posts: 339

Re: Formulas for the solution of Diophantine equations.

https://mathoverflow.net/questions/2647 … 193#275193

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.

individ

Member

Registered: 2014-03-16

Posts: 339

Re: Formulas for the solution of Diophantine equations.

https://mathoverflow.net/questions/2777 … -variables

Rewrite the system of equations.

The solution of this system can be written as.

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

individ

Member

Registered: 2014-03-16

Posts: 339

Re: Formulas for the solution of Diophantine equations.

https://math.stackexchange.com/question … uch-xyz22x

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.

individ

Member

Registered: 2014-03-16

Posts: 339

Re: Formulas for the solution of Diophantine equations.

https://mathoverflow.net/questions/2793 … 390#279390

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.

individ

Member

Registered: 2014-03-16

Posts: 339

Re: Formulas for the solution of Diophantine equations.

https://math.stackexchange.com/question … -relations

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

individ

Member

Registered: 2014-03-16

Posts: 339

Re: Formulas for the solution of Diophantine equations.

https://math.stackexchange.com/question … 54#2430154

Solution we write.

individ

Member

Registered: 2014-03-16

Posts: 339

Re: Formulas for the solution of Diophantine equations.

https://math.stackexchange.com/question … 78#2435778

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 

; 

;

individ

Member

Registered: 2014-03-16

Posts: 339

Re: Formulas for the solution of Diophantine equations.

https://math.stackexchange.com/question … 2-d2abx-s2

For the system of equations.

Make such a change.

The solution can be written as.

individ

Member

Registered: 2014-03-16

Posts: 339

Re: Formulas for the solution of Diophantine equations.

https://math.stackexchange.com/question … 98#2507298

For the equation.

The solution can be written directly.

individ

Member

Registered: 2014-03-16

Posts: 339

Re: Formulas for the solution of Diophantine equations.

https://math.stackexchange.com/question … 03#2568803

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.

individ

Member

Registered: 2014-03-16

Posts: 339

Re: Formulas for the solution of Diophantine equations.

https://math.stackexchange.com/question … 64#2596464

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

  are any integer.

individ

Member

Registered: 2014-03-16

Posts: 339

Re: Formulas for the solution of Diophantine equations.

https://math.stackexchange.com/question … 1-y2y1n2n1

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

.

individ

Member

Registered: 2014-03-16

Posts: 339

Re: Formulas for the solution of Diophantine equations.

http://math.hashcode.ru/questions/15654 … 576#156576

individ

Member

Registered: 2014-03-16

Posts: 339

Re: Formulas for the solution of Diophantine equations.

https://math.stackexchange.com/question … -b-i-3-and

individ

Member

Registered: 2014-03-16

Posts: 339

Re: Formulas for the solution of Diophantine equations.

https://math.stackexchange.com/question … l2-in-inte

You can record such a parameterization of solutions.