<![CDATA[Math Is Fun Forum / Formulas for the solution of Diophantine equations.]]> 2022-12-29T13:07:45Z FluxBB http://www.mathisfunforum.com/viewtopic.php?id=20642 <![CDATA[Re: Formulas for the solution of Diophantine equations.]]> https://math.stackexchange.com/question … -equations

There is a whole group of such systems of equations.

Such a system is solved as standard. First, we write down the parametrization of one equation.

And then we find the parameterization for the necessary parameters.

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<![CDATA[Re: Formulas for the solution of Diophantine equations.]]>

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<![CDATA[Re: Formulas for the solution of Diophantine equations.]]>

There was a question. When there are solutions

and

Each attempt at a solution gives a new formula for some reason. It didn't fit.

The following formula gives the necessary solutions.

The general formula should contain 3 more parameters.

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<![CDATA[Re: Formulas for the solution of Diophantine equations.]]> https://math.stackexchange.com/question … 16#4599816

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<![CDATA[Re: Formulas for the solution of Diophantine equations.]]> It turned out that parameterization of such a form can be made up a lot. For example, this. Which can save us from the symmetry of the formula. But so far it has been possible to obtain a formula with large coefficients...

If there are other formulas, then we must try to find one that is not symmetric and with small coefficients.

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<![CDATA[Re: Formulas for the solution of Diophantine equations.]]> https://math.stackexchange.com/question … -equations

You can write such a parametrization.

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<![CDATA[Re: Formulas for the solution of Diophantine equations.]]> http://math.hashcode.ru/questions/24950 … вное-число

Решений можно записать два. Одно когда взаимно простые....

Сумма их имеет вид.

И когда нет...

Ну и дальше всё совсем просто... взаимно простые когда скобки некоторые равны 1... ну или -1

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<![CDATA[Re: Formulas for the solution of Diophantine equations.]]> https://mathoverflow.net/questions/4320 … 221#432221

So the problem boils down to finding solutions to the general Pell equation. Then the problem is just in writing a convenient formula for describing solutions.
In addition to a convenient formula, there is another problem - through which Pell equations these solutions are described. Oddly enough, the usual Pell equation appears everywhere. Of this kind.

And there are two possible options for describing solutions. The first option is to use the standard Pell equation. And use such formulas, for example.

https://artofproblemsolving.com/community/c3046h1049910

https://artofproblemsolving.com/community/c3046h1048219

I don't particularly like these formulas. In order to use them, it is necessary to solve the Pell equation. Therefore, it was a question of using solutions of the general and standard Pell equations. That's strange. that it is always necessary to use the standard Pell equation to find solutions.

Since the square shape can always be reduced to the general Pell equation.

Using solutions of the standard Pell equation.

And using solutions of the general Pell equation.

You can write a formula for finding the next solution using the previous ones.

It was interesting, and if you don't do transformations, write an explicit formula. Will the same pattern persist in this case?

You still need a general equation.

And the standard equation.

Solutions can be written like this.

To obtain all solutions sequentially. They are usually used in the formula of the first solutions and going consistently to the big ones. Although formally, any solutions can be used in these formulas. For any solutions, the formula works.

The interesting thing is that no matter how the form of the general Pell equation changes. The form of solutions to this equation will be given by the standard Pell equation.

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<![CDATA[Re: Formulas for the solution of Diophantine equations.]]> https://math.stackexchange.com/question … 72#4370472

There are various variants of such equations. For example like this.

https://artofproblemsolving.com/communi … _variables

When solving such equations.

Pythagorean triples can help us. Take any two Pythagorean triples.

Then the solutions can be written in this form. Then the truth needs to be reduced by a common divisor.

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<![CDATA[Re: Formulas for the solution of Diophantine equations.]]> https://math.stackexchange.com/question … 86#4039186

To solve the Diophantine equation.

It is necessary to use the solution of the following equation.

There are solutions when the coefficient can be represented as the sum of squares.

And the solution itself can be presented in this form.

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<![CDATA[Re: Formulas for the solution of Diophantine equations.]]> https://mathoverflow.net/questions/8835 … 028#369028

https://arxiv.org/abs/2008.04440

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<![CDATA[Re: Formulas for the solution of Diophantine equations.]]> https://math.stackexchange.com/question … e-equation

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<![CDATA[Re: Formulas for the solution of Diophantine equations.]]> https://math.stackexchange.com/question … s-part-two

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