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individ

Member

Registered: 2014-03-16

Posts: 329

Re: Formulas for the solution of Diophantine equations.

https://math.stackexchange.com/question … 71#3214271

Use another equation.

And we use solutions to the Pell equation.

any number.

Decisions then write down so.

individ

Member

Registered: 2014-03-16

Posts: 329

Re: Formulas for the solution of Diophantine equations.

https://mathoverflow.net/questions/3343 … 624#334624

To solve the Diophantine equation.  Where

is any given number...

Solutions can be expressed through the following solutions to the Pell equation.

And then the decisions can be recorded...

individ

Member

Registered: 2014-03-16

Posts: 329

Re: Formulas for the solution of Diophantine equations.

https://math.stackexchange.com/question … y2-a2ab-b2

Solution of equation.

Can be written using solutions to the Pell equation....

Knowing the first solutions for....

The following can be found by the formula.

Will make a replacement for the simplified entry....

And then the solutions of the equation can be written in the form...

So the Pell equation has infinitely many solutions, it means for any number

- solutions will always be and infinitely many.

Last edited by individ (2019-07-03 00:21:35)

individ

Member

Registered: 2014-03-16

Posts: 329

Re: Formulas for the solution of Diophantine equations.

A link to the discussion of the equation....

https://dxdy.ru/topic134512.html

I kind of wrote 5 parametric solution.

For the case when 

or 

.... get the formula for 4 cubes...

Last edited by individ (2019-07-11 16:28:48)

individ

Member

Registered: 2014-03-16

Posts: 329

Re: Formulas for the solution of Diophantine equations.

https://math.stackexchange.com/question … s-part-two

Last edited by individ (2019-09-14 19:35:05)

individ

Member

Registered: 2014-03-16

Posts: 329

Re: Formulas for the solution of Diophantine equations.

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

individ

Member

Registered: 2014-03-16

Posts: 329

Re: Formulas for the solution of Diophantine equations.

https://mathoverflow.net/questions/8835 … 028#369028

https://twitter.com/republicofmath/stat … 5088661504

https://twitter.com/mathMGb/status/1293392350819913728

https://arxiv.org/abs/2008.04440

individ

Member

Registered: 2014-03-16

Posts: 329

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.

individ

Member

Registered: 2014-03-16

Posts: 329

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.