Formulas for the solution of Diophantine equations.

individ · 2014-03-16 18:18:06

Euler and not only he wanted so much to find these formulas. Time has passed to get them succeeded, but now they are not wanted.
Formula generally look like this: Do not like these formulas. But this does not mean that we should not draw them.

To start this equation zayimemsya, well then, and others.

Solutions can be written if even a single root.

, ,
Then the solution can be written.

In the case when the root

whole. Solutions have the form.

In the case when the root

whole. Solutions have the form.

Since these formulas are written in general terms, require a certain specificity calculations.If, after a permutation of the coefficients, no root is not an integer. You need to check whether there is an equivalent quadratic form in which, at least one root of a whole. Is usually sufficient to make the substitution

or more In fact, this reduces to determining the existence of solutions in certain Pell's equation. Of course with such an idea can solve more complex equations. If I will not disturb anybody, slowly formula will draw. number integers and set us.
I understand that these formulas do not like. And when they draw - or try to ignore or delete.
Formulas but there are no bad or good. They either are or they are not. In equation
integer coefficients which specify the conditions of the problem.
For a more compact notation, we introduce a replacement.

Then the formula in the general form is:

And more.

are integers and are given us. Since formulas are written in general terms, in the case where neither the root is not an integer, it is necessary to check whether there is such an equivalent quadratic form in which at least one root of a whole. If not, then the solution in integers of the equation have not.

Last edited by individ (2014-03-16 18:58:45)

individ · 2014-03-17 20:14:06

To show the possibility of the method. Take for example, solving systems of non-linear Diophantine equations.

The system of equations:

Solutions have the form:

If sitema little other equations:

Solutions can be written:

If the same formula can be written differently:

If we introduce the coefficients in the system:

Here is a partial solution when the root of the whole:

- Integers and sets us.

individ · 2014-03-18 06:28:05

That blocked me? Why?

bobbym · 2014-03-18 07:52:12

What blocked you?

individ · 2014-03-18 17:12:29

Strangely, I can not paint formula. Message appears that with this account does not accept replies.
Now try again.
Blocked sending messages. Had so cleverly use.

I draw these formulas have probably 8 months. They constantly remove and block. Here, for example if we take a system of linear equations is not. Invented the method of their decisions, but even on the forums I can not discuss them.
In this forum, they put http://www.mathforum.ru/forum/read/1/13491/page/2/
Comes to the ridiculous, these are my formula copied here: http://mathoverflow.net/questions/146768/solutions-of-system-of-diophantine-equations
Neither and others do not understand how to solve, but I'm still blocked. Okay, who wants to can see it. I'll draw the other redundant system.

In the system of equations:

Solutions have the form:

- integers and sets us.

set of equations:

has solutions

- integers and sets us.

Its theme I want to say that finding the solutions of Diophantine equations is the methods and ideas are great ideas from number theory. I do not make allegations and bring formula.
The strange thing is that these other ideas are extremely aggressive. Believe everything is known and does not need new ideas. But my above formulas show that there can come up with new ideas.

Last edited by individ (2014-03-18 18:01:01)

bobbym · 2014-03-19 11:53:36

Hi;

Blocked, in what way? Your first link is to a forum entirely in Russian. I know a smattering of that language but nowhere near enough to understand them or you.

The second link is the MO, I am a member there. Your ideas were looked at by some people there. They may be a little gruff and tough but they did look. And do not worry, I will get around to using my good friend M to check your equations.

individ · 2014-04-01 04:05:53

solutions Diophantine equation Madame Zarangesh.

You can record almost that trivial, but interesting is another case. When making formula takes some sort of a beautiful view. I only solutions are obtained using solutions of Pell: For example, if we use the solutions: Solutions have the form:

If we make the change:

Then:

If we use the solutions of Pell's equation:

And we use the substitution:

Then the solutions are of the form:

If the ratio of Pell's equation:

has the form

- what some integer number of any character.
We make the change:

Then the solution can be written:

Besides multiple solutions - when you come to solve the equations still to Pell's equation. It would be very interesting if you have an idea where it was possible to do without them. After substituting numbers for all primitive solutions will have to be divided into common divisor. Hopefully no need to explain?

bobbym · 2014-04-01 04:41:11

You are posting a lot of relationships, no proofs, no verification, no examples, no links and no replies to my comments.

In post #7

My first question is p a prime?

individ · 2014-04-01 05:43:11

Did not understand what question?
number

in the last report, there are solutions of Pell equations.
For example:
:

:

bobbym · 2014-04-01 10:20:12

Wanted to make sure. Normally a Pellian is denoted as x^2 - ny^2 = ± 1. p usually means a prime.

The first part of your last post appears sound and at least held up under an empirical attack.

individ · 2014-04-01 16:33:42

Why the topic moved to another location?
Nothing empirical there. All displayed strictly. Just do the calculations I still can not draw.
Do not like these formulas. Even on the forums constantly wash.

Agnishom · 2014-04-01 17:50:09

Why the topic moved to another location?

The 'Formulas' section is heavily moderated. We do not want anything confusing there

individ · 2014-04-01 18:11:29

Typically, solutions of Diophantine equations look quite cumbersome. There is nothing from me depends. Equations themselves decide which formula should be. It is due to the fact that some of their earlier cumbersome formula could not be obtained.

Agnishom · 2014-04-01 19:27:30

The administrator thinks your formulas are not clear enough or empirical or incorrect or something.

The forum software has some problem. It randomly blocks people. I've been blocked before as well. We are sorry

bobbym · 2014-04-01 19:33:37

Why the topic moved to another location?

Euler Avenue is for proven work. Your work as I have said comes with no proofs, no links to earlier material, no examples. Until it checks out then I will move it back.

Nothing empirical there.

There is also no proof provided of your methods. I had to check your method by trying lots of examples. I have only tried the last one.

Agnishom · 2014-04-01 19:35:28

All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

bobbym · 2014-04-01 19:41:10

A piece of work needs some sort of verification. If the author can not provide a proof it should be his responsibility to furnish a some strong evidence at least. This I have done for the last question.

Agnishom · 2014-04-01 19:54:26

You an verify this guy's work buy checking if it works for some Diophantine equations.

Where is the last question?

bobbym · 2014-04-01 19:56:54

The top part of post #7.

You an verify this guy's work buy checking if it works for some Diophantine equations.

Not exactly, but at least I failed to find a counterexample. All I can say is that it might be correct.

Agnishom · 2014-04-01 20:00:31

That should be a good "evidence".

bobbym wrote:

Proofs are for mathematics, Believing does not require proof.

individ · 2014-04-01 20:03:24

Formulas have been tested more than once and not only in one forum.
If you have questions I'll try to answer them.
Calculation method is new and has not been used, so the calculation methods can not show.
At this time, these formulas are trying to block and not to publish. So by the first guess about the method of calculation. But until they have obtained and I remained on the forums only these formulas draw.

bobbym · 2014-04-01 20:08:34

Formulas have been tested more than once and not only in one forum.

Testing formulas can give you some confidence in the result but it is not a proof.

At this time, these formulas are trying to block and not to publish.

I do not agree. No one has prevented you from posting your work. What they may have prevented you from doing is stating that they are correct.

Agnishom · 2014-04-01 20:18:20

I do not agree. No one has prevented you from posting your work.

Your forum software has. It is old and is Kaboobly Doo. I suggest you use phpBB or Vanilla immediately.

bobbym wrote:

Proofs are for mathematics, Believing does not require proof.

bobbym · 2014-04-01 20:26:21

Your forum software has. It is old and is Kaboobly Doo. I suggest you use phpBB or Vanilla immediately.

I do not think you and him are talking about the same thing.

individ · 2014-04-01 20:58:08

Nothing but a pen and paper, not used. All formulas were derived.
Fidelity formulas easily verified. Take equation is substituted for him and turns todzhestvo decision. Who do not believe can check.
While it is possible to use only such evidence

Math Is Fun Forum

#1 2014-03-16 18:18:06

Formulas for the solution of Diophantine equations.

#2 2014-03-17 20:14:06

Re: Formulas for the solution of Diophantine equations.

#3 2014-03-18 06:28:05

Re: Formulas for the solution of Diophantine equations.

#4 2014-03-18 07:52:12

Re: Formulas for the solution of Diophantine equations.

#5 2014-03-18 17:12:29

Re: Formulas for the solution of Diophantine equations.

#6 2014-03-19 11:53:36

Re: Formulas for the solution of Diophantine equations.

#7 2014-04-01 04:05:53

Re: Formulas for the solution of Diophantine equations.

#8 2014-04-01 04:41:11

Re: Formulas for the solution of Diophantine equations.

#9 2014-04-01 05:43:11

Re: Formulas for the solution of Diophantine equations.

#10 2014-04-01 10:20:12

Re: Formulas for the solution of Diophantine equations.

#11 2014-04-01 16:33:42

Re: Formulas for the solution of Diophantine equations.

#12 2014-04-01 17:50:09

Re: Formulas for the solution of Diophantine equations.

#13 2014-04-01 18:11:29

Re: Formulas for the solution of Diophantine equations.

#14 2014-04-01 19:27:30

Re: Formulas for the solution of Diophantine equations.

#15 2014-04-01 19:33:37

Re: Formulas for the solution of Diophantine equations.

#16 2014-04-01 19:35:28

Re: Formulas for the solution of Diophantine equations.

#17 2014-04-01 19:41:10

Re: Formulas for the solution of Diophantine equations.

#18 2014-04-01 19:54:26

Re: Formulas for the solution of Diophantine equations.

#19 2014-04-01 19:56:54

Re: Formulas for the solution of Diophantine equations.

#20 2014-04-01 20:00:31

Re: Formulas for the solution of Diophantine equations.

#21 2014-04-01 20:03:24

Re: Formulas for the solution of Diophantine equations.

#22 2014-04-01 20:08:34

Re: Formulas for the solution of Diophantine equations.

#23 2014-04-01 20:18:20

Re: Formulas for the solution of Diophantine equations.

#24 2014-04-01 20:26:21

Re: Formulas for the solution of Diophantine equations.

#25 2014-04-01 20:58:08

Re: Formulas for the solution of Diophantine equations.

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