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To solve the Diophantine equation.
It is necessary to use the solutions of the Pell equation.
Then the solutions can be written as follows.
For positive you need to take decisions at
.https://math.stackexchange.com/question … 64#2596464
The solution of the system of equations can be written in this form.
are any integer.https://math.stackexchange.com/question … 03#2568803
For this type of equations - which has something in common...
Record regarding
- A different equation. And replace solving a system of equations, the solution of the same equation.We write as....
And instead of solving the system, we solve it.
You can write a simple solution:
- any integer asked us.A similar method can be used for other type of equations.
https://math.stackexchange.com/question … 98#2507298
For the equation.
The solution can be written directly.
https://math.stackexchange.com/question … 2-d2abx-s2
For the system of equations.
Make such a change.
The solution can be written as.
https://math.stackexchange.com/question … 78#2435778
For the equation.
All variations of the same formula. As the number of solutions of course need to consider all possible factorization. Too much ends quickly. The number must be greater than 1.
Consider this example.
;Let
Means ; ;https://math.stackexchange.com/question … 54#2430154
Solution we write.
https://mathoverflow.net/questions/2793 … 390#279390
A better equation to solve in General.
As already mentioned, the task is reduced to some equivalent to the Pell equation. Actually reduced to this form.
Solution we write.
Or so.
https://math.stackexchange.com/question … uch-xyz22x
For the equation.
It is possible to reduce the parameterization of the solutions to some equivalent to the Pell equation.
It has the form.
These parameters can be recorded through the solution of the equation Pell.
https://mathoverflow.net/questions/2777 … -variables
Rewrite the system of equations.
The solution of this system can be written as.
https://mathoverflow.net/questions/2647 … 193#275193
There is one idea. To search for the solution of the equation.
If we know any solution
of this equation. Then it is possible to find another . Make such a change.Then the following solution can be found by the formula.
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Better system to write this.
For simplicity, the formula is written. Solution write using other parameters which will be determined next.
Express these options through the following polynomial.
Express these options through the following polynomial.
And now Express them via the parameters set by us
.The formula is very cumbersome and I wasn't able to test it. We must believe not made any mistake?
More simple parameterizations.
https://math.stackexchange.com/question … m3n3-holds
For the equation.
We use the solutions of the equation Pell.
Then the solution can write.
And again.
All the numbers can have any sign.
Hi can you suggest a good charting software, i am trying to plot conic section, intersection of cylindrical cones and a plane. If you can suggest a good charting software that would be great. I tried Asymptote and geogebra... not working out for me.
thanks
Do not write the question here! If you need to open a separate topic. There can be about a specific equation to tell.
I solve the equation. Programs do not write.
Build a curve at the intersection of two surfaces of a standard task. What is the program?
If you need to find the integral points. Formulate problem as a system of nonlinear equations. Then it will try to solve it.
https://math.stackexchange.com/question … 01#2280801
Spread by multipliers...
Decisions can be recorded.
https://math.stackexchange.com/question … x2y2-z2w21
Write solutions like this.
https://math.stackexchange.com/question … e-equation
For the solution of the equation.
Solution we write expanding on the multipliers.
And ;http://math.stackexchange.com/questions … 24#2214424
For the solution of the equation.
You must use the solutions of the equation.
You can use solutions which are recorded in the subject. http://math.stackexchange.com/questions … 219#709219
Then using the solutions of this equation can be substituted into the formula and find us.
http://math.stackexchange.com/questions … te-integer
For the equation.
You can write such a parameterization.
http://math.stackexchange.com/questions … rfect-squa
For odd numbers.
Decompose the number into factors.
The solution can be written as.
It is seen that for all odd numbers are infinitely many solutions, not just
.http://mathoverflow.net/questions/26315 … e-integers
Representation of a number we write.
I think that the only way to record the desired polynomial is to use the solutions of any equation.
Knowing the solutions of this equation and substituting them into the linear Diophantine equation.
variables which are solutions of this equation. Then the solution of the first equation can be written as.https://www.artofproblemsolving.com/com … umthory_13
Knowing one solution, others will find the formula.
Decisions will be.
http://math.stackexchange.com/questions … 09#2129909
For the system of equations.
Lay on multipliers.
Solutions written in this form. Any whole number. It is seen that solutions in integers there is not only for but for any other integer.This formula will be better ....
Decompose the number $T$ in two different ways.
http://math.stackexchange.com/questions … c-acb-sqrt
Solve the system.
The solution can be written as.