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It was necessary to write the solution in a more General form:
- integers.Decomposing on the factors as follows:
The solutions have the form:
For the equation:
The solution can be written using the factorization, as follows.
Then the solutions have the form:
I usually choose the number
such that the difference: was equal to:Although your desire you can choose other.
Solutions of the equation:
you can record if the root of the whole:
Then using the solutions of the equation Pell:
Then the formula of the solution, you can write:
If the root is a need to find out if this is equivalent to the quadratic form in which the root of the whole. This is usually accomplished this replacement:
in such numberForgot to say. The characters inside the brackets do not depend on the sign of the Pell equation. It depends only before
Equation:
Has the solution:
Equation:
Has the solution:
- Any integers.I think that it is necessary to adjust the formula solutions.
For the equation:
Solutions can be written.
For the system of equations:
Solutions have the form:
- integers of any sign.Solutions Pythagorean triples:
You can also submit through another Pythagorean triples:
Here is some formula, although they can write an infinite amount. All a matter of taste.
Or these:
Then using these triples can come to others.
- what some integers.For one, this entire kindergarten. Why is he interested in threes.
Strangely enough, the solution is finite.
for the equation:
If it is possible to decompose the coefficient as follows:
Then the solutions are of the form:
Thought the solution is determined by the equation Pell, but when calculating the sign was a mistake. There's no difference, but the amount should be. Therefore, the number of solutions of course.
Interested in the solution in general Diophantine equations of the form:
- what some integer.Solutions similar equations can be written.
Since this equation is easy, as it is quite symmetrical.
Such a solution can write.
And solutions can be written:
Whether there are any thoughts how to solve this equation?
At first I thought to use for solving Pell's equation, but I think that you can do without.
It was necessary to prove the existence of solutions for all
Can all make it easier to prove that 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.So there is always a solution.
Solutions of the equation:
Can be written without using Pell's equation:
- what some integer.The interesting thing is that the equation:
If we use the equation Pell:
Coefficient is defined as follows:
- integers asked us. Then the solutions are of the form:And another formula.
...........................................................................................................................................
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In the second formula should be chosen so Pell's equation and its solution so that the fraction decreased and turned to an integer.
I discovered this by accident.
Quite an interesting event. Very few people could imagine that this is possible.
Instead of all let's understand washable.
It is idiots!
Now the formula potter. Check it.
And it turned out pretty interesting consequence that although the equation
These site are mad!http://mathoverflow.net/
I solved the equation
the equation:
Has a solution:
Has a solution:
- integers asked us.Sometimes you have to deal with this equation:
- integer coefficients.I wrote below - to start a particular solution of Diophantine equations.
To do this, use the solutions of Pell's equation:
I turned solutions such.
And more:
Probably it is necessary to draw a formula for the solution in the general form:
In the equation:
Solutions can be written:
- what some integers.For the equation:
Write down the solution when the number can be factored as follows.
Then use the solution of Pell's equation:
Where the coefficient is given by:
- integers asked us.Then the solution can be written:
And more.
It's pretty old equations that are solved by Euler.
the equation:
If we use the solutions of Pell's equation:
Solutions can be written:
- We ask ourselves. While the formula and can be written differently.Solving the equation.
got some solutions, but still the question remains. Below are all the decisions or not?
And more.
In the system of equations:
Another solution can be written.
All three formulas derived me just describe all solutions of the system. I think the question can be considered closed.
The system of equations with given coefficients:
has the solutions:
- integers asked us.Okay. I think it is necessary to show the formula, in order to more constructive discussion went.
The system:
Solutions can be written as:
Or this:
All of numbers can be any character.In Equation:
If the ratio is factored so:
Then we use the solutions of Pell's equation:
where:
Then the solutions are of the form:
All of numbers can be any character.