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The system of equations.
Equivalent to the need to solve the following system of equations.
To ease calculations we change.
Then we need the number to obtain Pythagorean triples can be found by the formula.
At any stage of the computation can be divided into common divisor.
For generalization of the Pell equation question arises constantly. For example in this thread.
http://math.stackexchange.com/questions … 363#831363
So writes the Pell equation in General form.
If we know any solution of this equation.
If we use any solutions of the following equation Pell.
Then the following solution of the desired equation can be found by the formula.
We must remember that then can be reduced by a common factor.
This equation is a variation on this type of equations.
http://math.stackexchange.com/questions … e-patterns
If we consider the equation of a certain type.
The solutions can be written simply using the sequence. The next element which is obtained from the previous one.
If we use the first element of the sequence.
Then the formula will look like.
If we use the first element of the sequence.
Then the formula will look like.
Equation;
Decisions are not always there. To find out, I use too much. Decompose the multiplier number.
Finding the number $t$ and $k$, the solution writes:
This system of Diophantine equations was proposed. Form APMO 2000. 2.
Has this type:
One easy solution can be written as:
- integers which we ask.It is better to write such a decision.
- any integer.One of the solutions to the equation. To get infinite amount of different.
If you can imagine.
Then decisions can be recorded.
If you can imagine.
Then decisions can be recorded.
- integers which we ask.For the system of equations:
If you can decompose the coefficient multipliers as follows:
Their work squares:
Then decisions can be recorded.
You can add another simple option. If the ratio can be written as:
Then decisions can be recorded.
If you solve the system of equations:
When the standard approach solution and using a replacement.
Then the solution can be written as :
- integers which we ask.For the equation:
If you can represent numbers as:
This decision when the coefficients are related through the equation of Pell.
To simplify calculations we will make this change.
Then the solution can be written:
- integers which we ask.For the equation:
If the number
is the problem any, and is such as this:Then the solution can be written:
- integers which we are set.It's my decision.
How did you solve the other you can see there.
http://mathoverflow.net/questions/38354 … 180#196180
For such equations:
- you can specify any, then decisions can be recorded. - integers asked us.For the equation:
write the formula so that it was easier to go through. To facilitate calculations will make the replacement.
Then decisions can be recorded and they are.
- integers, which we ask.I thought as this task to generalize and use for any numbers. It turned out that you can do without calculations. For the system of equations:
Enough to factor the following number:
Using these numbers you can easily write the solution of this system of equations.
When he wrote the equation he meant probably that entry.
It turns out, this equation has a connection with the Pell equation:
For
it is necessary to use the first solution . For it is necessary to use the first solution . Knowing what the decision can be found on the following formula.Using the solutions of the Pell equation can be found when there are solutions.
Will make a replacement.
Then the solution can be written: - integers asked us. May be necessary, after all the calculations is to obtain a relatively simple solution, divided by the common divisor.Equation if we write in the General form:
If in this equation there any equivalent to a quadratic form in which the root is an integer.
Then there are solutions. They can be written by making the replacement.
Then decisions can be recorded and they are as follows:
***
***
***
- integers asked us.For the system of equations:
Rewrite all the same this solution to compare.
- coefficients specified by the problem statement. I was interested in other solutions when solutions are not multiples.What about this equation:
For this we need to use the solutions of the Pell equation.
To find solutions easily. Knowing the first solution
-Knowing one solution, the following can be found by the formula.
Knowing any solution and using it, you can find the solution to this equation by the formula.
Or different formula.
For the equation:
You can write the solution as:
Or different.
- Integers asked us. Not a lot not in a convenient form the recorded decision, but it is. Go to positive numbers is not difficult.When I decided this Diophantine equation, it became clear. If the coefficients are expressed as follows.
Where
When you can represent the coefficients in this form. Where
any number.For simplicity we make a replacement.
- integers asked us.Then the solution can be written in this form.
I was already glad that there are such factors when their values have infinitely many solutions. But it turned out that it turns out one mutually simple solution. All other solutions are multiples of him.
The solution of the equation:
If you use Pythagorean triple.
Then the formula for the solution of this equation can be written.
- any integer asked us.It all comes down to the Pell equation - as I said. Considering specifically the equation:
Decisions are determined such consistency.
Where the next value is determined using the previous one.
You start with numbers.
-Using these numbers, the solution can be written according to a formula.
If you use an initial
-Then the solutions are and are determined by formula.
As the sequence it is possible to write endlessly. Then the solutions of the equation, too, can be infinite.
If you use a sequence with the first element.
-If you use a sequence with the first element.
-Using this sequence can be different. On its basis with the first element.
Then decisions will be.
It is necessary to take into account that the number can have a different sign. -
For such equations:
Using the solutions of the Pell equation.
You can write the solution.
The system of equations:
Formulas you can write a lot, but will be limited to this. Will make a replacement.
The solution then is.
- integers.
I thought it possible to write a solution without using equations Pell.
If you make this change.
The result of such decision.
Where the number
- integers and set us. You may need after you get the numbers, divided by the common divisor.If the equations:
For ease of calculation, let's make a replacement.
Where
- integers asked us. You can use numbers if you reduce them to common divisor. Knowing these numbers can be written solutions.You can also write another formula, but it will look more bulky. It is necessary to consider that may need to be divided into common divisor.
on . on