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#301 2014-12-04 22:36:08

individ
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Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

For the equation:

- integers.  You can write solutions:

It is easy to see that we can cut.  But you can do different.

Ask any number and place it on the multipliers in the following way.

Then the other numbers are.

- any integer asked us.

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#302 2014-12-06 20:25:25

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,838
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Re: Formulas for the solution of Diophantine equations.

How'd I solve this:

where k and L are positive integers

Last edited by Agnishom (2014-12-06 20:25:51)


'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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#303 2014-12-06 20:49:23

individ
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Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

This Pell equation:   

Take the first solution 

   

Following looking at the formula.   

   

   

Then the solutions are of the form:   

   

Last edited by individ (2014-12-06 20:51:17)

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#304 2014-12-06 23:10:46

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,838
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Re: Formulas for the solution of Diophantine equations.

Where p and s are any integers?


'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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#305 2014-12-06 23:35:33

individ
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Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

No.
I showed the formula how to find them.
This formula is equivalent to the Pell equation.


Only then should   
  multiply by 3.

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#306 2014-12-06 23:40:06

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,838
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Re: Formulas for the solution of Diophantine equations.

Oh, thank you.


'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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#307 2014-12-10 00:48:28

individ
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Registered: 2014-03-16
Posts: 309

Re: Formulas for the solution of Diophantine equations.

Then for the equation:

You can  write many upgrades parameterization similar to the previous one.

- integers asked us.

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#308 2014-12-14 20:39:18

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

Re: Formulas for the solution of Diophantine equations.

For the equation:

The solution can be written using the Pell equation:

To find them easily. Knowing what one solution can be found on the following formula.

To begin to

with

To begin to

with

Then the formula of the solutions can be written.

The upper sign according to the decision of the Pell equation for

.

Lower

.

   - these numbers can have any sign.


Sierpinski in his book for the equation:

Meant this formula.

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#309 2014-12-16 21:01:20

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

Re: Formulas for the solution of Diophantine equations.

For the equation:

Solutions have the form:

For the equation:

Solutions have the form:

- integers asked us.

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#310 2014-12-17 21:10:01

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

Re: Formulas for the solution of Diophantine equations.

Maybe it is better to solve such a system the system of Diophantine equations:

Then the solution can be written.

- integers asked us.

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#311 2014-12-19 19:52:52

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

Re: Formulas for the solution of Diophantine equations.

It became interesting for the General case. When the difference is a square?

Write so equation:

If you use the solutions of the Pell equation.

Then decisions can be recorded.

- can be of any sign. So the difference will be equal.

Mean difference solutions of the square when the difference of the coefficients of the square.

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#312 2014-12-21 20:32:46

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

Re: Formulas for the solution of Diophantine equations.

This is a variation on Fermat's theorem.  The number of solutions is nite.  Will write a more General equation:

If 

- Will ask themselves.  Then the solution can be written;

- integers asked us.  You must consider that you can solve then reduce by common divisor.

Well here!
Today posted on his Blog 200 post.
http://www.artofproblemsolving.com/Foru … ?u=206450&

Each post is at least one formula!

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#313 2014-12-23 20:09:35

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

Re: Formulas for the solution of Diophantine equations.

For the equation:

Using the first number. 

  - 

Let's use these numbers. Which are the sequence.  The following is found using the previous value according to the formula.

Using the numbers

- you can find solutions on formulas.

***

So the formula looked compact, you can do the replacement.

Then the solutions are.

***

Or such replacement.

Then the solutions are.

***

Interestingly, all this variety of formulas give the same solution. So that one can restrict the upper formula. The rest of the formula was drawn to show what interesting patterns there.

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#314 2014-12-25 22:14:29

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

Re: Formulas for the solution of Diophantine equations.

In order to solve the equation:

Write the equation in General form.  Select the condition that the number of integer:

In order to record the decisions we make use of the solutions of the Pell equation of this type.

Numbers: 

- Ask us and can have any sign.  Knowing the solutions of the Pell equation we can write the solution of this equation as:

If

- not a whole. Find an equivalent replacement form 
on
  .  Then the equation becomes:

You will have to find out if the equation solutions.

This was easy to do because   

  - unknown.  You can of course choose another replacement.

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#315 2015-01-02 23:28:21

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

Re: Formulas for the solution of Diophantine equations.

For such equations, you can use the standard approach.
One approach is to use equations Pell.  For the beginning will talk about a more simple way.


For the equation: 

Solutions have the form.

For the equation: 

Solutions have the form.

For the equation:

Solutions have the form.

- integers asked us.

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#316 2015-01-03 20:05:44

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

Re: Formulas for the solution of Diophantine equations.

Now will show you a different approach using the equations Pell.

For the equation:

Will set the number

- and can be of any sign.  Then use the Pell equation.

Where:

Then the solutions are.

And another solution:

Be aware that if the ratio of the Pell equation

- fold the square, can be reduced. In the formula, too, should be reduced.

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#317 2015-01-04 21:14:29

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

Re: Formulas for the solution of Diophantine equations.

For the equation:

Decisions can be recorded.

- integers asked us.

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#318 2015-01-06 23:10:44

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

Re: Formulas for the solution of Diophantine equations.

Thought it possible to simplify in order to be able to write the solutions of the equation. For this we use the decomposition of the number $c$ on the multipliers.

To record decisions have to know first the solution of the Pell equation

.

And solving the following equation Pell

.

Then the formula is as follows.

The problem in finding the first solution for General Pell equation

.

The meaning of the solution is that to factor the number.

Then degradable factoring the difference. 

If the following expression may be a square. 

Then the first solution is written simply.

Such record these formulas will greatly simplify the calculations. Always better to have a formula.

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#319 2015-01-08 22:25:06

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

Re: Formulas for the solution of Diophantine equations.

If the equations:

For ease of calculation, let's make a replacement.

Where 

- integers asked us.  You can use numbers if you reduce them to common divisor.  Knowing these numbers can be written solutions.

You can also write another formula, but it will look more bulky. It is necessary to consider that may need to be divided into common divisor.

on
.
on 

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#320 2015-01-11 19:33:14

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

Re: Formulas for the solution of Diophantine equations.

I thought it possible to write a solution without using equations Pell.

If you make this change.

The result of such decision.

Where the number

- integers and set us.  You may need after you get the numbers, divided by the common divisor.

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#321 2015-01-15 21:27:05

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

Re: Formulas for the solution of Diophantine equations.

The system of equations:   

 

Formulas you can write a lot, but will be limited to this. Will make a replacement. 

 

 

 

The solution then is.

 

 

 

- integers.

Last edited by individ (2015-01-15 21:28:22)

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#322 2015-01-16 19:41:39

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

Re: Formulas for the solution of Diophantine equations.

For such equations:

Using the solutions of the Pell equation.

You can write the solution.

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#323 2015-01-17 18:10:32

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

Re: Formulas for the solution of Diophantine equations.

It all comes down to the Pell equation - as I said.  Considering specifically the equation:   

 

Decisions are determined such consistency. 
Where the next value is determined using the previous one.   

 

You start with numbers. 

-
   

Using these numbers, the solution can be written according to a formula.   

 

 

If you use an initial

-
 

Then the solutions are and are determined by formula. 

 

As the sequence it is possible to write endlessly. Then the solutions of the equation, too, can be infinite.

If you use a sequence with the first element.

-

Then decisions can be recorded.

If you use a sequence with the first element.

-

Using this sequence can be different. On its basis with the first element.

-

Then decisions will be.

It is necessary to take into account that the number can have a different sign. -

Last edited by individ (2015-02-20 17:47:29)

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#324 2015-01-19 18:16:27

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

Re: Formulas for the solution of Diophantine equations.

The solution of the equation:

If you use Pythagorean triple.

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

- any integer asked us.

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#325 2015-01-21 20:14:26

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

Re: Formulas for the solution of Diophantine equations.

When I decided this Diophantine equation, it became clear. If the coefficients are expressed as follows.

Where 

 

When you can represent the coefficients in this form.  Where

any number.

For simplicity we make a replacement.

- integers asked us.

Then the solution can be written in this form.

I was already glad that there are such factors when their values have infinitely many solutions.  But it turned out that it turns out one mutually simple solution. All other solutions are multiples of him.

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