Formulas for the solution of Diophantine equations.

individ · 2014-10-25 19:01:17

Did not understand. Decisions will be infinitely many.

Agnishom · 2014-10-25 19:17:25

Agnishom wrote:

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/

The question asks to enumerate all the solution in those bounds

individ · 2014-10-25 19:25:08

I strive to ensure that people have used my formulas in the calculations.
To understand their use greatly simplifies the calculations.
If I write the numbers on the formula, no one will pay attention.

So. The formula I wrote the rest consider themselves.
If not satisfied then it is not necessary.

Agnishom · 2014-10-25 20:10:02

Ok, I will use your formula in my calculations. Thank you

individ · 2014-10-27 21:26:01

I think it is better to solve this system of equations, and solutions to find the right settings.

Decisions can be recorded.

Or so:

Then the number found as:

- integers asked us.

Last edited by individ (2014-10-27 21:28:24)

individ · 2014-10-28 20:35:36

For the system of equations:

If you can the amount be represented as a sum of squares:

Then the solution can be written:

- integers asked us.

individ · 2014-11-01 01:19:12

I decided on a different way to write the solution. These formulas do not give all solutions.
For the equation:

If we choose this number "

" so it was odd and it was not a multiple of 3.

Then for and another solution:

Lay on multipliers and find

.

Then

It is necessary that "

" was greater than zero. Otherwise there will be confusion.

individ · 2014-11-01 17:52:05

This system of equations:

Solutions have the form:

- integers asked us.

individ · 2014-11-02 18:36:17

For the equation:

You can record a simple solution.

Where

- an odd number.
The number - select the appropriate parity so that the bracket was even.

individ · 2014-11-05 17:51:25

I think for the equation:

It is necessary to record decisions. We will use the solutions of the Pell equation.

And then the solutions are of the form:

individ · 2014-11-08 01:50:21

The simple formula for the equation:

You can write this:

- integers asked us.

If there is a one simple solution, it will be necessary to reduce to the corresponding

.

There are other formulas. But they are bulky. Don't know whether it makes sense to write them.

individ · 2014-11-08 21:49:35

For the equation:

Will make a replacement that formula was compact.

- integers asked us. Then decisions can be recorded.

individ · 2014-11-10 18:27:42

For the equation:

Can specify any number :

. Then decisions will be.

- any integer asked us.

individ · 2014-11-14 18:25:05

The system of equations.

Solutions have the form:

- integers asked us.

Agnishom · 2014-11-14 23:16:12

If you do not mind me asking, what does 'integers asked us' mean?

individ · 2014-11-14 23:48:23

Integers which we ask. Any.
We would like.
It's not my fault that I don't understand. Google translates so.

individ · 2014-11-15 23:11:18

If we consider the sum of two triangular numbers, that they were square.

If you use the solutions of the Pell equation:

Using the solutions of this equation find the solution of another equation Pell:

Using the solutions of these two equations, we can write the solution as:

- Can have any sign.

individ · 2014-11-17 01:41:25

Probably will be more evident if we write this equation:

For this we need to use the solutions of the Pell equation:

Then the solutions are of the form:

***

And accordingly:

From the last formula can be seen as if not asked number $t,k$ always come to Diophantine equation 4th or more from a single variable. Or two variables. Depending on how you want to solve it.

As Tue proved such equations if you have solution then the number of solutions of course.

individ · 2014-11-17 23:57:49

For the system of equations:

Solutions have the form.

- integers asked us.

individ · 2014-11-18 22:55:00

When the number of variable in the system of Diophantine equations a little. The number of solutions of course means the formula of solutions it is impossible to give. I think we should write some beautiful system. You can for example put the following task: when consecutive squares have a certain look.

For example: sequential squares representable as sums of squares.

Quite simply, the decision to write. Ask these numbers -

. That the following sum is a multiple of 4 or odd.

Lay on multipliers and find -

. Solutions have the form:

You can consider another combination.

You also can specify the number of -

. And to factor.

Find the number -

. And we write the solutions of the equation as:

You can also write a formula connecting these numbers, but it is bulky. This approach is more simple.
The same standard approach can be used if you want to choose another polynomial form.

individ · 2014-11-23 01:11:42

If there was a need for a simpler entry to the equation:

Then write a simple solution using any two integers -

.

In this form it is easier and for all coefficients.

individ · 2014-11-25 21:54:13

I think not to introduce additional equations, and directly solve the system of equations.

Using integer parameters

- Will make a replacement.

Then the solution can be written as:

To obtain relatively Prime solutions - after substitution should be reduced to common divisor.

This problem was there.
It reduced the formula - so give the link to the topic.
http://math.stackexchange.com/questions … 7x-37-dots

individ · 2014-11-29 04:02:42

For such equations :

You can write the formula.

individ · 2014-11-29 22:40:38

The system of equations:

Solutions have the form:

- integers asked us.

individ · 2014-12-01 18:43:23

For the equation:

The solution can be written:

- any integer.

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#276 2014-10-25 19:01:17

Re: Formulas for the solution of Diophantine equations.

#277 2014-10-25 19:17:25

Re: Formulas for the solution of Diophantine equations.

#278 2014-10-25 19:25:08

Re: Formulas for the solution of Diophantine equations.

#279 2014-10-25 20:10:02

Re: Formulas for the solution of Diophantine equations.

#280 2014-10-27 21:26:01

Re: Formulas for the solution of Diophantine equations.

#281 2014-10-28 20:35:36

Re: Formulas for the solution of Diophantine equations.

#282 2014-11-01 01:19:12

Re: Formulas for the solution of Diophantine equations.

#283 2014-11-01 17:52:05

Re: Formulas for the solution of Diophantine equations.

#284 2014-11-02 18:36:17

Re: Formulas for the solution of Diophantine equations.

#285 2014-11-05 17:51:25

Re: Formulas for the solution of Diophantine equations.

#286 2014-11-08 01:50:21

Re: Formulas for the solution of Diophantine equations.

#287 2014-11-08 21:49:35

Re: Formulas for the solution of Diophantine equations.

#288 2014-11-10 18:27:42

Re: Formulas for the solution of Diophantine equations.

#289 2014-11-14 18:25:05

Re: Formulas for the solution of Diophantine equations.

#290 2014-11-14 23:16:12

Re: Formulas for the solution of Diophantine equations.

#291 2014-11-14 23:48:23

Re: Formulas for the solution of Diophantine equations.

#292 2014-11-15 23:11:18

Re: Formulas for the solution of Diophantine equations.

#293 2014-11-17 01:41:25

Re: Formulas for the solution of Diophantine equations.

#294 2014-11-17 23:57:49

Re: Formulas for the solution of Diophantine equations.

#295 2014-11-18 22:55:00

Re: Formulas for the solution of Diophantine equations.

#296 2014-11-23 01:11:42

Re: Formulas for the solution of Diophantine equations.

#297 2014-11-25 21:54:13

Re: Formulas for the solution of Diophantine equations.

#298 2014-11-29 04:02:42

Re: Formulas for the solution of Diophantine equations.

#299 2014-11-29 22:40:38

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#300 2014-12-01 18:43:23

Re: Formulas for the solution of Diophantine equations.

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