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**individ****Member**- Registered: 2014-03-16
- Posts: 329

http://math.stackexchange.com/questions … -and-z-x-y

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.

*Last edited by individ (2016-01-20 20:36:28)*

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**individ****Member**- Registered: 2014-03-16
- Posts: 329

Use the ideas that basically involve formal and General approach.

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.

*Last edited by individ (2016-03-21 17:11:50)*

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**individ****Member**- Registered: 2014-03-16
- Posts: 329

http://math.stackexchange.com/questions … ated-lemma

To describe the solutions of the equation.

I think best would be to describe a solution using

parameters.*Last edited by individ (2021-07-03 02:04:50)*

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**individ****Member**- Registered: 2014-03-16
- Posts: 329

The task from there.

http://www.artofproblemsolving.com/comm … ber_theory

Solution always can be written, for example.

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**individ****Member**- Registered: 2014-03-16
- Posts: 329

One binary form.

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.

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**individ****Member**- Registered: 2014-03-16
- Posts: 329

The equation from there

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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**individ****Member**- Registered: 2014-03-16
- Posts: 329

http://math.stackexchange.com/questions … divide-b28

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.

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**individ****Member**- Registered: 2014-03-16
- Posts: 329

http://math.stackexchange.com/questions … 65#1826965

For the system of equations .

The solution can be written in this form.

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**individ****Member**- Registered: 2014-03-16
- Posts: 329

http://mathoverflow.net/questions/24373 … 774#243774

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.

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**individ****Member**- Registered: 2014-03-16
- Posts: 329

http://mathoverflow.net/questions/25017 … nal-number

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 ;

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**individ****Member**- Registered: 2014-03-16
- Posts: 329

http://math.stackexchange.com/questions … 50#1965550

For Diophantine equation.

You can record a parameterization.

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**individ****Member**- Registered: 2014-03-16
- Posts: 329

http://mathoverflow.net/questions/25326 … al-surface

For such systems of equations.

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

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**individ****Member**- Registered: 2014-03-16
- Posts: 329

https://twitter.com/SeanReeves/status/8 … 1787171840

Using the solutions of the equation Pell.

Solution write.

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**individ****Member**- Registered: 2014-03-16
- Posts: 329

http://math.hashcode.ru/questions/11382 … 1%82%D1%8B

Solution.

*Last edited by individ (2016-12-05 01:52:09)*

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**individ****Member**- Registered: 2014-03-16
- Posts: 329

http://www.artofproblemsolving.com/comm … mo_problem

Solutions have the form.

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**individ****Member**- Registered: 2014-03-16
- Posts: 329

http://math.stackexchange.com/questions … 26#2069326

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.

*Last edited by individ (2016-12-22 19:47:55)*

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**individ****Member**- Registered: 2014-03-16
- Posts: 329

http://math.stackexchange.com/questions … 83#2070383

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.

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**individ****Member**- Registered: 2014-03-16
- Posts: 329

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**individ****Member**- Registered: 2014-03-16
- Posts: 329

http://math.stackexchange.com/questions … 30#2077030

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.

*Last edited by individ (2017-01-14 00:50:00)*

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**individ****Member**- Registered: 2014-03-16
- Posts: 329

http://mymathforum.com/number-theory/33 … -bz-c.html

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.

*Last edited by individ (2017-01-08 20:19:04)*

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**individ****Member**- Registered: 2014-03-16
- Posts: 329

http://math.stackexchange.com/questions … m21-and-m1

To solve this system of equations - it is necessary to solve the system.

It is necessary to find a parameterization to figurirovallo Pell. It is possible for example to record this.

We need a case of when.

Knowing the first decision

The rest can be found by the formula.

Although this equation can be not enough. We need to find when there are multiple solutions.

*Last edited by individ (2017-01-18 05:07:35)*

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**individ****Member**- Registered: 2014-03-16
- Posts: 329

http://math.stackexchange.com/questions … c-acb-sqrt

Solve the system.

The solution can be written as.

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**individ****Member**- Registered: 2014-03-16
- Posts: 329

http://math.stackexchange.com/questions … 09#2129909

For the system of equations.

Lay on multipliers.

Solutions written in this form. Any whole number. It is seen that solutions in integers there is not only for but for any other integer.This formula will be better ....

Decompose the number $T$ in two different ways.

*Last edited by individ (2017-03-03 21:10:02)*

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**individ****Member**- Registered: 2014-03-16
- Posts: 329

https://www.artofproblemsolving.com/com … umthory_13

Knowing one solution, others will find the formula.

Decisions will be.

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**individ****Member**- Registered: 2014-03-16
- Posts: 329

http://mathoverflow.net/questions/26315 … e-integers

Representation of a number we write.

I think that the only way to record the desired polynomial is to use the solutions of any equation.

Knowing the solutions of this equation and substituting them into the linear Diophantine equation.

variables which are solutions of this equation. Then the solution of the first equation can be written as.Offline