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https://math.stackexchange.com/question … -equations

There is a whole group of such systems of equations.

Such a system is solved as standard. First, we write down the parametrization of one equation.

And then we find the parameterization for the necessary parameters.

There was a question. When there are solutions

andEach attempt at a solution gives a new formula for some reason. It didn't fit.

The following formula gives the necessary solutions.

The general formula should contain 3 more parameters.

It turned out that parameterization of such a form can be made up a lot. For example, this. Which can save us from the symmetry of the formula. But so far it has been possible to obtain a formula with large coefficients...

If there are other formulas, then we must try to find one that is not symmetric and with small coefficients.

https://math.stackexchange.com/question … -equations

You can write such a parametrization.

http://math.hashcode.ru/questions/24950 … вное-число

Решений можно записать два. Одно когда взаимно простые....

Сумма их имеет вид.

И когда нет...

Ну и дальше всё совсем просто... взаимно простые когда скобки некоторые равны 1... ну или -1

https://mathoverflow.net/questions/4320 … 221#432221

So the problem boils down to finding solutions to the general Pell equation. Then the problem is just in writing a convenient formula for describing solutions.

In addition to a convenient formula, there is another problem - through which Pell equations these solutions are described. Oddly enough, the usual Pell equation appears everywhere. Of this kind.

And there are two possible options for describing solutions. The first option is to use the standard Pell equation. And use such formulas, for example.

https://artofproblemsolving.com/community/c3046h1049910

https://artofproblemsolving.com/community/c3046h1048219

I don't particularly like these formulas. In order to use them, it is necessary to solve the Pell equation. Therefore, it was a question of using solutions of the general and standard Pell equations. That's strange. that it is always necessary to use the standard Pell equation to find solutions.

Since the square shape can always be reduced to the general Pell equation.

Using solutions of the standard Pell equation.

And using solutions of the general Pell equation.

You can write a formula for finding the next solution using the previous ones.

It was interesting, and if you don't do transformations, write an explicit formula. Will the same pattern persist in this case?

You still need a general equation.

And the standard equation.

Solutions can be written like this.

To obtain all solutions sequentially. They are usually used in the formula of the first solutions and going consistently to the big ones. Although formally, any solutions can be used in these formulas. For any solutions, the formula works.

The interesting thing is that no matter how the form of the general Pell equation changes. The form of solutions to this equation will be given by the standard Pell equation.

https://math.stackexchange.com/question … 72#4370472

There are various variants of such equations. For example like this.

https://artofproblemsolving.com/communi … _variables

When solving such equations.

Pythagorean triples can help us. Take any two Pythagorean triples.

Then the solutions can be written in this form. Then the truth needs to be reduced by a common divisor.

What is NFT?

I think you can start by reading the Wikipedia article. https://en.wikipedia.org/wiki/Non-fungible_token

But the meaning is simple. This is not a mutually replaceable token. It is unique and one of a kind. Blockchain technology is a tool that confirms that it was created by this person and the information posted was created by him.

The creation of a file with information is associated with the wallet number. And the owner of this token is the owner of the wallet.

Anyone can view the information that is embedded in the token. Anyone can download the token itself. It costs nothing. This is a plus. Information can be distributed absolutely free of charge. Simply, the owner of the token is a certain wallet.

Sure.... it may be that someone jumps the token and then wants to register it under their own name, but delete the entire blockchain chain. To lose information about who was the owner - he can not. It will be very easy to prove plagiarism.

The formula is generally a very tenacious subject. You can force it to be deleted. You can force it to be ignored, but over time there will definitely be a question about its origin. And then the blockchain technology will tell you about the priority.

I publish formulas on the forums. I give advice and tell many people. And often the ways in which the information goes are difficult to understand. From time to time, the formula pops up somewhere. And this is a good mechanism that will let you know who came up with the formula.

Many people have no other choice. This is all made up from no source.

No one will allow me to publish an article in the journal. There is no chance that the magazine will want to take the article. At the same time, I do not want the idea to be lost. And the formulas themselves disappeared. So that then in 100 years someone rediscovered them again.

And if on the Internet, these formulas will live their own life... then maybe it will bring some benefit.

Some abbreviations are standard and well-known to everyone. If someone does not understand some of my words , then I will explain. Ask me a question.

Don't worry about someone selling something. Technology that is created for enrichment... It will be used by us to create knowledge and disseminate information absolutely free of charge. It's all free and doesn't cost a cent.... just don't tell the owners of the site!

I just came up with this new idea.... apply so tokens. Let's try to apply it and see what happens!

As I said earlier... very often, the new is unusual and people try to fight with the new. The man himself is conservative. And the new one is likely to refuse.

I proposed a new mechanism. Of course, it will be worth changing over time. Not on the same site to post information, and probably create some separately, but so far there is no other option-you have to use what is there.

So I posted a topic on this forum...

https://www.mathisfunforum.com/viewtopic.php?id=20642

Or on another forum I post a topic. For example, I like these equations... others are different. But not everyone likes them, right?

Many people, for example, are categorically against what I do. And they struggle with it. But I have the result. There are some formulas... there is some information. Yes... everything at this point in time is considered bad.

But some people like it. For example, I solve and collect solutions of Diophantine equations. The other can collect rows or magic squares.

And the subject of the collection will be the solution of the equation or what that idea is. This will lead to the fact that people themselves will apply forces to increase knowledge.

Many are unable to publish their results... I have for example on very many forums formulas generally disappeared. This is just another mechanism for preserving knowledge and sharing it.

In magazines, to get access to an article, you need to pay money, and here anyone can download and view it for free.

Technology changes a lot. And in this case, technology can give an impetus to the popularization of science. Anyone can try to solve a previously unsolved problem. After creating a token, it will announce its solution to the problem.

This will preserve the copyright and give others the opportunity to review and evaluate. There is no critical discussion of the results in science right now.

This is just one of the tools for popularizing science. Let people start solving equations - how to draw cats!

**individ**- Replies: 5

One of the biggest problems in science is when an idea comes up. This is to convey it. Tell people about it. And on the other hand, it is access to information.

In general, it is extremely difficult for people to convey their ideas. A huge problem in publications. Modern scientific journalism is built in such a way that it is virtually impossible to publish an article in a scientific journal-if you do not have working contacts with the editorial board.

But this does not mean that discoveries are made only at Harvard. A lot of new things are being learned by different people all over the world. And even if a person does not work at the University - this does not mean that he will not be able to discover something new.

Knowledge is such a subject-which is not a shame to get from any place.

Limiting the ability to communicate your ideas to others - this often leads to the need for others to come to the same result.... only after some time.

There is still a problem in the assessment of knowledge. Often new things can't make their way - because most people don't like the idea. I always don't like new things.

And information, even if someone doesn't like it. Must be saved. After passing the test of time - it will be clear whether it is valuable or not.

On the other hand, there is a problem with access to information. Not only is it published very subjectively, but scientific journals have severely restricted access to it. Just reading some article costs from $ 50.

Much is said about these problems, but in fact nothing is being done for the better. So I decided to come up with a new mechanism for the author's interaction with the audience.

And the NFT will help us with this. Now all the work with them is in the field of art. Everyone draws pictures, but I think that science has a place there. The value of the NFT is determined by what? Rarity, exclusivity, made by some respected author, novelty. But much more valuable-when it is difficult to get it. When you need to spend a lot of effort to create something.

And what could be better than nature? The effort that can be expended to find out what the phenomenon, the law... solve a problem or equation. Come up with an algorithm or a way to simplify the task. These efforts can be very complex and large.

And that's when you make a discovery. You solve an equation or a problem. You come up with something new. That is, any idea. And then you form it in the form of an NFT. This will allow you to retain the priority right to the idea itself. There is an opportunity to work in the field of patents. On the other hand, you tell everyone about your idea. Anyone can access and learn the information. At the same time, as the owner of the token, you will have the right to be called the person who invented it. And at the same time, the value of the token will be evaluated. If a university cooperates with a scientist, it will buy the token from him. This way, research and development will be funded, and the more tokens a University has and the greater their value, the better it is.

This will stimulate the work of scientists. On the other hand, it will allow sponsors to finance exactly those areas that they are interested in.

And the value of such a token will be greater. Although there are infinitely many equations and problems in nature, not all of them can be solved. The first person to solve a previously unsolved problem, problem, or equation will issue their token. If someone is interested in this, they can always check both the creation time and the information. This will facilitate the dissemination of information and maintain priority.

So I decided to start. So far, I have created 217 tokens for the formula for solving Diophantine equations. The formulas were not previously known, I received them myself and could not publish them in the journal. Now they are released in the form of NFT.

The token collection is located at this address.

https://opensea.io/collection/a-real-street-cat-3

https://math.stackexchange.com/question … 86#4039186

To solve the Diophantine equation.

It is necessary to use the solution of the following equation.

There are solutions when the coefficient can be represented as the sum of squares.

And the solution itself can be presented in this form.

A link to the discussion of the equation....

https://dxdy.ru/topic134512.html

I kind of wrote 5 parametric solution.

For the case when

or .... get the formula for 4 cubes...https://math.stackexchange.com/question … y2-a2ab-b2

Solution of equation.

Can be written using solutions to the Pell equation....

Knowing the first solutions for....

The following can be found by the formula.

Will make a replacement for the simplified entry....

And then the solutions of the equation can be written in the form...

So the Pell equation has infinitely many solutions, it means for any number - solutions will always be and infinitely many.

https://mathoverflow.net/questions/3343 … 624#334624

To solve the Diophantine equation. Where

is any given number...Solutions can be expressed through the following solutions to the Pell equation.

And then the decisions can be recorded...

https://math.stackexchange.com/question … 71#3214271

Use another equation.

And we use solutions to the Pell equation.

any number.Decisions then write down so.

https://math.stackexchange.com/question … l2-in-inte

You can record such a parameterization of solutions.