Math Is Fun Forum

http://mymathforum.com/number-theory/33 … -bz-c.html

For Diophantine equation

If the root is integer.   

We use the solutions of the equation Pell. 

The solution then can be written in this form.

To find all solutions is necessary to solve the more General equation. With different coefficients.

http://math.stackexchange.com/questions … al-x2p2-p2

For such equations.

Solution write.

http://math.stackexchange.com/questions … 26#2069326

For the system of equations.

Solutions can be parameterized.

It is interesting that such triples can be too much. The formula can be increased to any number. That is the same to write not only for 4 partitions, but for any number. The main thing that all the variables were not identical to each other.

http://math.hashcode.ru/questions/11382 … 1%82%D1%8B

  Solution.

http://mathoverflow.net/questions/25326 … al-surface
For such systems of equations.

It is better to use an algebraic approach. He gives at once rational decisions.

http://mathoverflow.net/questions/25017 … nal-number

For finding solutions in the case if the square is rational.

Use the same formula. For finding all solutions need to consider all possible solutions of the equation Pell.  Consider the case where these solutions exist.  To do this, lay on the multiplier coefficients.

And you need to check when this Pell equation has the solutions?

And then the solution is substituted in the above formula.

Consider the case which invited.

;

;

;

;

;

;

Use the first solution. 

  ; 

Then ;

http://math.stackexchange.com/questions … 65#1826965

For the system of equations .

The solution can be written in this form.

The equation from there
http://math.stackexchange.com/questions … n-integers

For the equation.

You can write the solution in this form.

Such a record is better because allows to solve the symmetrical equation with any number of summands. It is only necessary to increase the number of parameters.

The task from there.
http://www.artofproblemsolving.com/comm … ber_theory

Solution always can be written, for example.

Use the ideas that basically involve formal and General approach.
But apparently the time has not come for them. Even attempt a discussion and a proposal to solve the equations in General form leads to the opposition.
http://mathoverflow.net/questions/23414 … -equations
Attempt to offer to solve an equation of the form

Only causes aggression and a desire to remove.  With this notation explicitly perceived very well.

It is seen that the time for the ideas of formalization and a common approach is not yet. 
Although attempts to reduce such ideas and methods you need to do.

http://mathoverflow.net/questions/88220 … ct-squares

I think it is better to solve a more formal task.  We write the system.

We need to find solutions

- that was an arithmetic progression.  This will help the solution of the equation Pell.

Knowing any solution of the equation Pell

you can always find the following formula.

Having any decision - can immediately write down the solution of this system.

Interesting here is that the

looks like an arithmetic progression.

http://math.stackexchange.com/questions … y2z2-xyz22

For the equation.

If we use the solutions of Pell equations.

Using my replacement.

Decisions will be.