Formulas for the solution of Diophantine equations.

individ · 2015-08-20 03:22:22

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)

individ · 2015-08-21 05:35:38

The system of book 2: objectives 24, 25.

The solutions can be written as.

Or so.

individ · 2015-08-29 23:29:30

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)

individ · 2015-09-01 19:38:11

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)

individ · 2015-09-04 18:47:24

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

Wrote the system.

The solutions can be written as.

Any integers.

individ · 2015-09-09 19:30:12

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

You can record such a decision.

any integers.

individ · 2015-09-12 00:48:29

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)

individ · 2015-10-08 22:04:40

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.

individ · 2015-10-29 20:38:08

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

Agnishom · 2015-10-30 03:58:57

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

individ · 2015-10-30 04:28:20

I am not interested in his opinion and questions!

Agnishom · 2015-10-31 03:19:35

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.

individ · 2015-10-31 18:14:00

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

Decision on the record.

Or.

individ · 2015-11-11 22:51:20

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.

individ · 2015-11-12 18:53:52

For the equation.

You can record such decisions.

individ · 2015-11-15 19:28:33

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.

individ · 2015-11-25 21:33:17

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.

individ · 2015-12-09 00:48:55

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

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

individ · 2015-12-11 03:22:41

Considered this equation.

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

individ · 2015-12-12 23:04:27

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.

individ · 2015-12-14 20:29:14

For the equation.

A more compact solution.

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

individ · 2015-12-24 21:48:22

For the equation.

You can write a fairly simple formula.

individ · 2016-01-08 03:55:54

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

For the equation.

If we use the solutions of Pell equations.

Using my replacement.

Decisions will be.

individ · 2016-01-10 21:35:22

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.

individ · 2016-01-12 01:48:37

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.

Math Is Fun Forum

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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