Formulas for the solution of Diophantine equations.

individ · 2014-07-29 22:42:51

Perhaps you should write a more General solution of the equation.

If you ask any integers:

- and use the solutions of the equation Pell.

The solution, you can write:

You can write a simple solution without the Pell equation.

If:

individ · 2014-07-30 00:38:17

In General formula generic for Pythagorean triples looks a little different.

If the number can be represented as a sum of squares.

The solution has the form:

individ · 2014-08-01 20:25:58

One said that this equation has no solutions. Because the equation Farm has no solutions. It turned out that the solution of this equation is.

Fermat's theorem cannot be proved, if you do not know how to solve Diophantine equations.

This approach cannot be used. This equation has a solution.

The solutions have the form:

- any integer.

individ · 2014-08-01 22:56:43

You can write another solution. This is equivalent to the equation:

The solutions have the form:

................................................

..............................................

Will make a replacement.

The solution has the form:

individ · 2014-08-04 22:03:33

Then this equation. If there is a solution - they are infinitely many.

Finding solutions-it factor. To factor.

We use the solutions of the equation Pell.

Then the solutions are.

individ · 2014-08-06 03:08:25

Equation:

You can write a simple solution:

- any integer asked us.

individ · 2014-08-09 21:28:19

Equation:

If the ratio is the square.

Using the solutions of the equation Pell.

Then the solutions are.

individ · 2014-08-13 03:34:16

In the equation.

Solutions are provided by the Pell equation.

And have a look.

Solving the Pell equation can be found. Knowing the past can be found .

You can start with.

All numbers can have any sign.
Another can be reduced to 4. And come to the equation.

individ · 2014-08-14 21:30:54

The system of Diophantine equations:

Solutions have the form:

An interesting case when:

For this we need to solve the Pell equation:

And solutions to substitute in the formula.

individ · 2014-08-15 01:04:27

For the system of equations:

- choose an integer, so that the bracket was intact.

Then the solutions are.

individ · 2014-08-16 02:45:24

Do there exist four distinct integers such that the sum of any two of them is a perfect square?

This is equivalent to solving the following system of equations:

Let:

- any asked us integers.

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

Then the solutions are of the form:

individ · 2014-08-16 06:18:58

This system was solved and there.

http://www.artofproblemsolving.com/Foru … 6&t=602478

You can compare solutions.

individ · 2014-08-17 17:40:51

On Fildovsky award nominated Manjul Bhargava.

http://www.mathunion.org/general/prizes/2014/

His work there.

http://arxiv.org/pdf/1006.1002v2.pdf

http://arxiv.org/pdf/1007.0052v1.pdf

Funny. He can't solve a single equation. Can't write a single formula.
Even says on the contrary that the formulas cannot be obtained.
But it's not. In some cases, to obtain a formula for the solution.

They strongly opposed the formulas.

bobbym · 2014-08-17 22:51:23

Fildovsky award? What is that?

individ · 2014-08-17 23:33:52

There you can see.
http://www.mathunion.org/general/prizes/2014/

As not fair. He no single formula did not write and say it is good work.
I wrote so many formulas and are not allowed to publish.

bobbym · 2014-08-17 23:53:15

He is a Fields medal winner.

individ · 2014-08-19 22:55:11

Equation:

Using the solutions of the Pell equation:

Solutions have the form:

individ · 2014-08-23 21:29:46

Equation:

Has a solution:

- any integer.

individ · 2014-08-24 00:41:34

Equation:

Has a solution:

- any integer.

Last edited by individ (2014-08-24 02:52:05)

individ · 2014-09-01 20:50:03

Equation:

Decisions will be :

...............

................

.................

If you use the solutions of the Pell equation:

And the following substitutions:

Or:

Then the solutions are of the form:

Last edited by individ (2014-09-01 20:58:33)

individ · 2014-09-01 20:53:39

Forgot in the last formula, we need to 4 be reduced. -

Already corrected and reduced.

Last edited by individ (2014-09-01 21:00:00)

individ · 2014-09-05 03:10:48

You can write such a formula.

individ · 2014-09-11 18:27:20

As I said, for 8 unknown parameters and the formula goes bulky.

3 - the formula looks like this: http://math.stackexchange.com/questions … 527#738527

Will consider here the special case when:

Then the solutions are of the form:

- integers asked us. It is clear that if you will satisfy the condition we can always write such a simple solution. It is easy enough to see how it turns out.

individ · 2014-09-18 02:52:29

It is necessary to say a few words about the formula about which I have spoken. For the equation:

Need to write this simple formula.

I think that this formula gives all solutions. Mutually simple solution obtained after reduction to common divisor.

For example there was a similar situation with the equation:

It is enough to write the formula generates an endless series of decisions in all degrees. For this we use the Pythagorean triple.

And the number of their sets.

- what some integers. Then the solution can be written.

And mutually simple solutions can get if you cut down on common divisor. Although there will be not one simple solution.

individ · 2014-09-19 19:35:07

It is clear that in equation:

Decisions are determined by the solutions of the Pell equation:

We need to write a formula that clearly shows what was the substitution desired. Will make a replacement.

Then the solutions are of the form:

It is necessary to consider another solution. In known solutions

- to find their counterparts. Upon substitution into the formula they give solutions. Are they using the formula.

We must be careful that the signs not to confuse them.

The Pell Equation:

Is very simple. And for the first solution:

;

You can find the rest by the formula:

; - any previous solution of the Pell equation.

Math Is Fun Forum

#226 2014-07-29 22:42:51

Re: Formulas for the solution of Diophantine equations.

#227 2014-07-30 00:38:17

Re: Formulas for the solution of Diophantine equations.

#228 2014-08-01 20:25:58

Re: Formulas for the solution of Diophantine equations.

#229 2014-08-01 22:56:43

Re: Formulas for the solution of Diophantine equations.

#230 2014-08-04 22:03:33

Re: Formulas for the solution of Diophantine equations.

#231 2014-08-06 03:08:25

Re: Formulas for the solution of Diophantine equations.

#232 2014-08-09 21:28:19

Re: Formulas for the solution of Diophantine equations.

#233 2014-08-13 03:34:16

Re: Formulas for the solution of Diophantine equations.

#234 2014-08-14 21:30:54

Re: Formulas for the solution of Diophantine equations.

#235 2014-08-15 01:04:27

Re: Formulas for the solution of Diophantine equations.

#236 2014-08-16 02:45:24

Re: Formulas for the solution of Diophantine equations.

#237 2014-08-16 06:18:58

Re: Formulas for the solution of Diophantine equations.

#238 2014-08-17 17:40:51

Re: Formulas for the solution of Diophantine equations.

#239 2014-08-17 22:51:23

Re: Formulas for the solution of Diophantine equations.

#240 2014-08-17 23:33:52

Re: Formulas for the solution of Diophantine equations.

#241 2014-08-17 23:53:15

Re: Formulas for the solution of Diophantine equations.

#242 2014-08-19 22:55:11

Re: Formulas for the solution of Diophantine equations.

#243 2014-08-23 21:29:46

Re: Formulas for the solution of Diophantine equations.

#244 2014-08-24 00:41:34

Re: Formulas for the solution of Diophantine equations.

#245 2014-09-01 20:50:03

Re: Formulas for the solution of Diophantine equations.

#246 2014-09-01 20:53:39

Re: Formulas for the solution of Diophantine equations.

#247 2014-09-05 03:10:48

Re: Formulas for the solution of Diophantine equations.

#248 2014-09-11 18:27:20

Re: Formulas for the solution of Diophantine equations.

#249 2014-09-18 02:52:29

Re: Formulas for the solution of Diophantine equations.

#250 2014-09-19 19:35:07

Re: Formulas for the solution of Diophantine equations.

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