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#251 2014-09-20 21:18:10

individ
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Registered: 2014-03-16
Posts: 339

Re: Formulas for the solution of Diophantine equations.

For the equation:

If parameter is specified

- the solution seems cumbersome, but if you find you only need:

Then the solution has the form:

- asked us integer corresponding parity.

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#252 2014-09-21 18:37:28

individ
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Registered: 2014-03-16
Posts: 339

Re: Formulas for the solution of Diophantine equations.

The task is quite simple. Taken from this thread.
http://www.artofproblemsolving.com/Foru … 7&t=607094

At first when I started to solve the equation of thought that you can specify only one factor.

Were you can ask any ratio

And the solution of the equation:

- any integer.

In order to attract attention to the method we have to solve a simple equation.

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#253 2014-09-24 18:32:52

individ
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Registered: 2014-03-16
Posts: 339

Re: Formulas for the solution of Diophantine equations.

Well, the formula itself Geronova triangle. 

 

If:

  -integers asked us. Then the solutions are. 

 

 

 

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#254 2014-09-24 23:10:21

individ
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Registered: 2014-03-16
Posts: 339

Re: Formulas for the solution of Diophantine equations.

One particular solution.

I have already said, where the formula in General.

For the equation:

If you use the solutions of the Pell equation.

And we have a number $y,n$ known. Moreover, any sign.

Then:

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#255 2014-09-29 03:48:42

individ
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Registered: 2014-03-16
Posts: 339

Re: Formulas for the solution of Diophantine equations.

For the equation:

This solution will .

- integers of any sign.

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#256 2014-10-02 00:31:04

individ
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Registered: 2014-03-16
Posts: 339

Re: Formulas for the solution of Diophantine equations.

I think that this method of calculation it is necessary to separately draw.

As I have repeatedly said formula in General looks pretty bulky.  And still remain questions about the completeness of the solution. So I decided that solutions should be found a little differently.

In Diofantos equation:

Put some numbers:   

Decompose to factor the following expression: 

Then we can define the following numbers:

Next, you can specify the desired number:

Subject to the following expression for the multiplier:

This will allow us to unambiguously identify numbers:

And for the full solution will be found by the formula two other numbers.

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#257 2014-10-02 19:07:23

individ
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Registered: 2014-03-16
Posts: 339

Re: Formulas for the solution of Diophantine equations.

For the equation:

You can write for example this solution:

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#258 2014-10-09 17:41:16

individ
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Registered: 2014-03-16
Posts: 339

Re: Formulas for the solution of Diophantine equations.

Equation:

Solutions have the form:

Solutions have the form:

- integers asked us any sign.

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#259 2014-10-10 23:32:51

individ
Member
Registered: 2014-03-16
Posts: 339

Re: Formulas for the solution of Diophantine equations.

Although the equation is simple. But it often occurs in many cases. Required to represent the sum of the squares of the product of the multipliers. And the inverse problem. To represent the number as a sum.

Therefore, it is convenient equation to represent and solve it like this:

Then the solution can be represented as:

Or so:

- integers of any sign.

It is necessary to consider not mutually simple solutions. Or Vice versa. Reduce to obtain relatively simple solutions.

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#260 2014-10-11 18:26:44

individ
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Registered: 2014-03-16
Posts: 339

Re: Formulas for the solution of Diophantine equations.

For the system of equations:

You can record solutions:

- integers asked us.

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#261 2014-10-13 00:50:33

individ
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Registered: 2014-03-16
Posts: 339

Re: Formulas for the solution of Diophantine equations.

Bad.
So much worked and solved systems of equations.
All gathered in this thread.
http://math.stackexchange.com/questions … -equations

And now. It's all removed.

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#262 2014-10-13 02:32:31

individ
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Registered: 2014-03-16
Posts: 339

Re: Formulas for the solution of Diophantine equations.

For not a lot of other systems of equations:

Solutions have the form:

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#263 2014-10-17 23:25:36

individ
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Registered: 2014-03-16
Posts: 339

Re: Formulas for the solution of Diophantine equations.

Review and generalize the well-known formula.
https://en.wikipedia.org/wiki/Brahmagupta%27s_formula 
To begin, write the formula of the solution of the following equation: 

 

Formulas of the solutions can be written. 

 

 

 

 

The square will be equal to: 

 

And the perimeter. 

 

- Integers asked us.
And now about the idea of solving the equation:   

 

- the integer specified by the problem statement. To do this, put this number on the multiplier. And we will solve the linear equation in the unknown
.   

 

The number

determine from the equation. 

Last edited by individ (2014-10-17 23:28:32)

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#264 2014-10-18 21:54:12

individ
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Registered: 2014-03-16
Posts: 339

Re: Formulas for the solution of Diophantine equations.

I became entangled in the problem and decided another equation.  It is necessary to solve this equation:

Solutions will be:

integers asked us.  Then the area of the quadrilateral are equal.

And its perimeter is equal to:

To solve the equation:

When the number of $F$ is set for the problem. Come to the need to solve the following equation:

Left to think. What is the easiest way to solve this equation.

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#265 2014-10-20 01:38:43

individ
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Registered: 2014-03-16
Posts: 339

Re: Formulas for the solution of Diophantine equations.

For the equation: 

 

You can write this simple solution: 

 

 

 

The ratio is given for the problem. 
integers asked us.

Last edited by individ (2014-10-20 01:39:40)

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#266 2014-10-22 04:26:12

individ
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Registered: 2014-03-16
Posts: 339

Re: Formulas for the solution of Diophantine equations.

For the equation:   

   If you choose any odd number "

"   

And lay multipliers.   

   

This formula will find

   

Then a fairly simple solution, you can write:

   

   

   

   

It is necessary that

was greater than zero. Otherwise there will be confusion.

Last edited by individ (2014-10-22 04:30:05)

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#267 2014-10-22 23:41:02

individ
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Registered: 2014-03-16
Posts: 339

Re: Formulas for the solution of Diophantine equations.

For this purpose it is necessary to solve the equation:

 

If the next root whole 

then it will be possible for this simple case record decisions:

 

 

 

 

 

 

 

-  integers asked us.

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#268 2014-10-23 20:21:43

individ
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Registered: 2014-03-16
Posts: 339

Re: Formulas for the solution of Diophantine equations.

We need to write generally speaking the more General equation:

Although I formula solutions recorded, but I see it is of interest expression solutions using any one of the known solution.

If we know what any one solution:

  - then you can write a formula for the solutions of this equation.

- any integer asked us.

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#269 2014-10-23 21:35:47

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: Formulas for the solution of Diophantine equations.

Hi individ;

May I request your help on a number theory problem?

Last edited by Agnishom (2014-10-23 21:39:16)


'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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#270 2014-10-23 21:45:34

individ
Member
Registered: 2014-03-16
Posts: 339

Re: Formulas for the solution of Diophantine equations.

Did not understand.
What?

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#271 2014-10-23 22:13:10

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: Formulas for the solution of Diophantine equations.

I appreciate your aptitude in number theory.

I was wondering if you can help me understand this question:

https://brilliant.org/community-problem … ot-a-typo/


'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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#272 2014-10-23 23:02:23

individ
Member
Registered: 2014-03-16
Posts: 339

Re: Formulas for the solution of Diophantine equations.

The question is not as simple as it seems. 
This task is equivalent to the system of equations:

 

It is necessary to solve the system of equations.  And to find out whether there is a solution

?

I think it would be an interesting case when you find out at what rate decisions will be. Very sorry. When I have time I will try to solve the system and the formula here will draw.

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#273 2014-10-23 23:25:13

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: Formulas for the solution of Diophantine equations.

It's okay.

Do you have any idea how I can get better at number theory?


'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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#274 2014-10-25 18:11:14

individ
Member
Registered: 2014-03-16
Posts: 339

Re: Formulas for the solution of Diophantine equations.

For systems of equations:   

   

Directly solved quite difficult because at first I found the solution of simple equations. And the decision recorded, so it didn't look bulky.   

   

You can write this solution:   

   

   

   

   

Number

- we will find as the solution of binary quadratic forms.   

   

The formula for the solutions of this equation there:   http://www.artofproblemsolving.com/blog/101140   

The number

- in the formula substituted such that the root was a rational number.

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#275 2014-10-25 18:47:50

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: Formulas for the solution of Diophantine equations.

Will that list out all solutions?


'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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