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#351 2015-08-20 03:22:22

individ
Member
Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

The task from there.

http://mathoverflow.net/questions/21519 … -mathbb-zt

If the number

is set by the problem statement. Then in the equation.

The solutions can be written as.

***

Symmetric solution to the previous one.


any integers.

Last edited by individ (2015-08-20 07:57:12)

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#352 2015-08-21 05:35:38

individ
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Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

The system of book 2: objectives 24, 25.

The solutions can be written as.

Or so.

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#353 2015-08-29 23:29:30

individ
Member
Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

The system was solved there.
http://math.stackexchange.com/questions … al-numbers

So it is easier to solve the system of equations is presented in this form.

Then the solution can be written as.

  Any integers.

The solution can be written in this form.

Last edited by individ (2015-08-30 21:03:30)

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#354 2015-09-01 19:38:11

individ
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Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

Problem is taken from there.
http://math.stackexchange.com/questions … 72#1418072

For the equation:

Use the standard approach of using Pell equations.  If you use solutions of this equation.

Decisions can be recorded.

If this equation Pell.

Then decisions can be recorded.

Or.

For this.

Number 

  can have any sign.

An interesting case of when. 

The formulas can be seen that while the

can be anything. And 

Last edited by individ (2015-09-01 19:40:31)

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#355 2015-09-04 18:47:24

individ
Member
Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

The task from there.
http://www.artofproblemsolving.com/comm … _solutions

Wrote the system.

The solutions can be written as.

  Any integers.

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#356 2015-09-09 19:30:12

individ
Member
Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

2 books of Diophantus task 34 , 35.  For the system of Diophantine equations.

You can record such a decision.

any integers.

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#357 2015-09-12 00:48:29

individ
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Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

I think this system of Diophantine equations.

It is better to solve in General. This notation allows to find solutions for any values 

. Solution we write better.

***

***

***

***

***

***

***

***

  integers which we can ask.

If we want to find out what the odds should be for example

.  Just substitute the formula and solve the equation. Everything will be reduced to the factorization.

Last edited by individ (2015-09-12 00:52:09)

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#358 2015-10-08 22:04:40

individ
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Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

The task from there.

http://math.stackexchange.com/questions … 18#1471718

For such a system.

Here is the result.

Number 

are selected so that the desired number intact.

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#359 2015-10-29 20:38:08

individ
Member
Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

For the equation is possible to write General standard formula - moreover it is symmetric.  Interestingly, so monjo to record for any number of summands.

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#360 2015-10-30 03:58:57

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,838
Website

Re: Formulas for the solution of Diophantine equations.

individ wrote:

The task from there.

http://mathoverflow.net/questions/21519 … -mathbb-zt

If the number

is set by the problem statement. Then in the equation.

The solutions can be written as.

***

Symmetric solution to the previous one.


any integers.

Igor Rivin has a point

4        
Would you care to elaborate on how you get these? – Igor Rivin Aug 20 at 15:43


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#361 2015-10-30 04:28:20

individ
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Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

I am not interested in his opinion and questions!

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#362 2015-10-31 03:19:35

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,838
Website

Re: Formulas for the solution of Diophantine equations.

If that is the case, I have nothing to say.

I urge you to reconsider that, anyway. When you post results without derivations in a cryptic way, they are not actually helping anybody and is more like talking to yourself.


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#363 2015-10-31 18:14:00

individ
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Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

Task found there. http://dxdy.ru/topic102317.html

Decision on the record.

Or.

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#364 2015-11-11 22:51:20

individ
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Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

http://math.stackexchange.com/questions … -both-prim

Show how the General formula allows us to solve this equation.

We write this equation differently.

Will do the replacement. 

Then the equation takes the form.

Now you can use the General formula. http://www.artofproblemsolving.com/comm … 46h1048219

Let the root equal to 1.  This means that using the solutions of the equation Pell.

Knowing the first solution.

You can find the following solution to the formula.

Using the General formula and the solution of the equation Pell.

You can write the solution of the equation in this form.

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#365 2015-11-12 18:53:52

individ
Member
Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

For the equation.

You can record such decisions.

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#366 2015-11-15 19:28:33

individ
Member
Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

The equation from there.
http://math.stackexchange.com/questions … -variables

For the equation:

You can record such decisions.

Or.

For the equation.

You can record such decisions.

Or.

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#367 2015-11-25 21:33:17

individ
Member
Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

http://math.stackexchange.com/questions … 69#1547069

For the equation.

If you know any solution

of this equation.  Then the formula for the solution of the equation can be written immediately.

  any integers.



For the equation.

You can set some numbers infinitely different way.

Then decisions can be recorded.

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#368 2015-12-09 00:48:55

individ
Member
Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

More interesting the other decisions - when the number is positive.

- the number is specified by the problem statement and can be any.

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#369 2015-12-11 03:22:41

individ
Member
Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

Considered this equation.

And to have found such a parameterization. Write can be useful.

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#370 2015-12-12 23:04:27

individ
Member
Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

The same formal record.  If we look for a parameterization not in 2 and in 3 option the problem can be solved quite simply.

For the equation.

If you know any solution

of this equation.  Then the formula for the solution of the equation can be written immediately.

-   any integers.

It is seen that such formulas can be written infinitely many.  If so like to use the well-known decision, it is better to write like this.  This will allow to solve the equation.

It is easy to see that there are solutions for any

. Ask yourself the number
. And place into factors.

Then the coefficient is set to

us so.

Finding all the factors of

you can write a formula for the parameterization of the solution of this equation.

The transition to 3 parameters

-   Allows not to use too much, but to write formulas.

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#371 2015-12-14 20:29:14

individ
Member
Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

For the equation.

A more compact solution.

Last edited by individ (2015-12-15 21:55:07)

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#372 2015-12-24 21:48:22

individ
Member
Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

For the equation.

You can write a fairly simple formula.

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#373 2016-01-08 03:55:54

individ
Member
Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

http://math.stackexchange.com/questions … y2z2-xyz22

For the equation.

If we use the solutions of Pell equations.

Using my replacement.

Decisions will be.

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#374 2016-01-10 21:35:22

individ
Member
Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

http://math.stackexchange.com/questions … an-squares

It is better to use a more General approach. We write the system.

If the number

, lay at the multipliers. We find then the desired settings.

Then the solution can be written as.

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#375 2016-01-12 01:48:37

individ
Member
Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

http://mathoverflow.net/questions/88220 … ct-squares

I think it is better to solve a more formal task.  We write the system.

We need to find solutions

- that was an arithmetic progression.  This will help the solution of the equation Pell.

Knowing any solution of the equation Pell

you can always find the following formula.

Having any decision - can immediately write down the solution of this system.

Interesting here is that the

looks like an arithmetic progression.

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