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**individ****Member**- Registered: 2014-03-16
- Posts: 310

Although equation:

Solutions have the form:

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Hi;

That formula is correct.

This can be simplified to

now using

and

When expanded out

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**individ****Member**- Registered: 2014-03-16
- Posts: 310

Yes, I know. I was just joking. I wanted to see someone reading my formula or not.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

I read all the posts. Occasionally I miss one but not often.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**individ****Member**- Registered: 2014-03-16
- Posts: 310

For example in the equation:

If we use the solutions of Pell's equation:

Then the solution can be written, where the numbers

whole and sets us.More.

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**individ****Member**- Registered: 2014-03-16
- Posts: 310

It is necessary then to just write another equation:

If we use the solutions of Pell's equation:

formula for the solution can be written, where

integers asked us.And more.

Interestingly, and if you add three triangular numbers, the formula would look like?

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**individ****Member**- Registered: 2014-03-16
- Posts: 310

If the equation:

We write the solution if the number of:

This can be represented as ideal:Then use the equation Pell:

- integers asked us.Then the solution can be written:

Or again:

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**individ****Member**- Registered: 2014-03-16
- Posts: 310

When solved this system of equations, it is still the question remains. The following formula describes all written decision or not? Of course you can write a formula in a different form, but it is interesting is not it, but what if there is enough of this formula?

Solutions are defined as integers.

To simplify the notation we make the change :

Then the formula becomes:

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**individ****Member**- Registered: 2014-03-16
- Posts: 310

For the equation:

We can write the solution:

And more:

- integers asked us.Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 310

equations

Can be expressed by integers . where the number of characterizes the degree of primitiveness.And more.

If we decide to factor

For a compact notation we replace :

then:

And the most beautiful solution. If we use the solutions of Pell's equation:

by the way

May appear as a factor in the decision and.Then the solutions are of the form::

If we change the sign :

Then the solutions are of the form:

Another solution of the equation:

- integers asked us.To facilitate the calculations we make the change.

If the ratio is as follows :

If the ratio is as follows:

Then the solutions are of the form:

Then the solutions are of the form:

If the ratio is as follows :

If the ratio is as follows

Then the solutions are of the form :

Then the solutions are of the form :

*Last edited by individ (2016-01-19 17:06:10)*

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Hi;

Did you check user17762's answer?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**individ****Member**- Registered: 2014-03-16
- Posts: 310

Hi! I do not understand what to check? And from whom?

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**individ****Member**- Registered: 2014-03-16
- Posts: 310

I get it. Learned that you had in mind. So he just wrote the number, and I am writing formulas. The funny thing is that many people write tsiferki and everyone says - it's good. I write the formula, and I scolded.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

He did not write just numbers,it is a Bezout type formula. Using any primitive solution he can get all others.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**individ****Member**- Registered: 2014-03-16
- Posts: 310

That's the difference between my method and from yours. You bust find a solution, and it will build the following solutions.

I decide equation. And get the formula. I search and guessing is not necessary.

Yes this approach I get a broader picture. More clarity.

I have a more interesting formulas are obtained. And there are such formulas on kotryh nobody commemorated.

So what to do? Delete them and forget? Or still think about where they came from?

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

I only called attention to it for reference. This is not a contest about whose work is the best. The first poster over there mentions Cohen's book. Do you have it?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**individ****Member**- Registered: 2014-03-16
- Posts: 310

I just read on the Russian. Such a book is not seen. My calculation method differs from existing methods.

If method good opportunities - this can be judged by the diversity of equations that decided.

You can see them in my blog. http://www.artofproblemsolving.com/Foru … p?u=206450

*Last edited by individ (2015-02-15 18:13:55)*

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

I write the formula, and I scolded.

No one is scolding you over there but they might ignore your work. There are reasons for that.

1) You seem to have a bias against finding a solution and then finding all others from it.

You bust find a solution, and it will build the following solutions.

How hard is it to find 1,2 and 3 as a solution? So the Bezout type formula has merit and is not inferior just because there is some guessing involved.

2) Math types, like the ones that you are posting to like to see proofs of statements. Your work has none.

3) They like to see references as to where the work comes from. Yours again has none of that.

I am not judging whether you are right or wrong just why no one is looking at it.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**individ****Member**- Registered: 2014-03-16
- Posts: 310

The first time when I was there appeared washed my formula.

I'm trying to prove that my method also has the right to exist, but I can not even publish the formula.

I always say that there is one correct method. This Diophantine geometry and everything that does not fit in it should be erased.

I even invented a method for solving systems of nonlinear algebraic equations. And what is the result? In Russia I was on all the forums is erased or blocked. I can not even on the forums just to paint formula.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

I always say that there is one correct method. This Diophantine geometry and everything that does not fit in it should be erased.

I do not agree, there are many methods to do everything.

Mathematicians like simplicity. I like speed. Your methods are not the simplest and not the fastest. Your methods have a right to exist but they are difficult to compute and tough to see how they are derived.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**individ****Member**- Registered: 2014-03-16
- Posts: 310

Who told you that?

I'm still on the method of calculation is not mentioned.

Formulas are obtained by long and beautiful, but the method itself is simple.

In fact, these formulas give the unexpected decision of another famous problem.

It appears there is an algebraic problem where to find a solution much easier than to check the correctness of the decision.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Who told you that?

Which part?

Formulas are obtained by long and beautiful, but the method itself is simple.

That was already stated as a problem. What method? You do not explain anything.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**individ****Member**- Registered: 2014-03-16
- Posts: 310

Do you want that on the forum, I spoke about the method of calculation - which was not published anywhere?

I always say that tell me about another method to erase. But the value of the method is much more - than the ability to draw on our formula.

So we'll have to wait.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

In post #60, do they not have 3 shorter parametrizations of that in Cohen's book?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**individ****Member**- Registered: 2014-03-16
- Posts: 310

I said that I do not have this book so I can not judge his formulas.

If these formulas you do not like - time for you to find others. But when solving the equations do not always come short. And it's not my fault - the equations themselves decide what they need to have a formula.

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