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Pretty old problem. Find such pentagonal numbers - which can be represented as a sum of two squares. That is, to prove that such numbers are infinitely many.
So I'm wondering - if I write a formula describing their decision whether it will be sufficient evidence that such numbers are infinitely many?
That is, the following Diophantine equation:
If we use the equation Pell:
Then using the solutions of this equation can be written solutions of the equation. Sign in and sign Pell alignment of the first alignment should be the same.
more:
Can I assume that the problem was solved? Proved - there are infinitely many pentagonal numbers which can be represented as the sum of two squares?
If I understand correctly, it is not my fault. Google translates it so. Probably better than the formula used to discuss them.
Here it is not my fault that the formulas are so long. Such they should be. There's nothing I can not do.
While we can be limited by the fact that to know there are formulas for some equations.
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
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.
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.
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.
Did not understand what question?
number
solutions Diophantine equation Madame Zarangesh.
If we make the change:
Then:
If we use the solutions of Pell's equation:
Then the solutions are of the form:
If the ratio of Pell's equation:
has the form- what some integer number of any character.
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?
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.
That blocked me? Why?
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.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.
, ,
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.
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.