Formulas for the solution of Diophantine equations.

individ · 2014-09-20 21:18:10

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.

individ · 2014-09-21 18:37:28

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.

individ · 2014-09-24 18:32:52

Well, the formula itself Geronova triangle.

If:

-integers asked us. Then the solutions are.

individ · 2014-09-24 23:10:21

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:

individ · 2014-09-29 03:48:42

For the equation:

This solution will .

- integers of any sign.

individ · 2014-10-02 00:31:04

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.

individ · 2014-10-02 19:07:23

For the equation:

You can write for example this solution:

individ · 2014-10-09 17:41:16

Equation:

Solutions have the form:

- integers asked us any sign.

individ · 2014-10-10 23:32:51

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.

individ · 2014-10-11 18:26:44

For the system of equations:

You can record solutions:

- integers asked us.

individ · 2014-10-13 00:50:33

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.

individ · 2014-10-13 02:32:31

For not a lot of other systems of equations:

Solutions have the form:

individ · 2014-10-17 23:25:36

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)

individ · 2014-10-18 21:54:12

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.

individ · 2014-10-20 01:38:43

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)

individ · 2014-10-22 04:26:12

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)

individ · 2014-10-22 23:41:02

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.

individ · 2014-10-23 20:21:43

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.

Agnishom · 2014-10-23 21:35:47

Hi individ;

May I request your help on a number theory problem?

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

individ · 2014-10-23 21:45:34

Did not understand.
What?

Agnishom · 2014-10-23 22:13:10

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/

individ · 2014-10-23 23:02:23

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.

Agnishom · 2014-10-23 23:25:13

It's okay.

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

individ · 2014-10-25 18:11:14

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.

Agnishom · 2014-10-25 18:47:50

Will that list out all solutions?

Math Is Fun Forum

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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

Re: Formulas for the solution of Diophantine equations.

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