Math Is Fun Forum

I think that this method of calculation it is necessary to separately draw.

As I have repeatedly said formula in General looks pretty bulky.  And still remain questions about the completeness of the solution. So I decided that solutions should be found a little differently.

In Diofantos equation:

Put some numbers:   

Decompose to factor the following expression: 

Then we can define the following numbers:

Next, you can specify the desired number:

Subject to the following expression for the multiplier:

This will allow us to unambiguously identify numbers:

And for the full solution will be found by the formula two other numbers.

One particular solution.

I have already said, where the formula in General.

For the equation:

If you use the solutions of the Pell equation.

And we have a number $y,n$ known. Moreover, any sign.

Then:

The task is quite simple. Taken from this thread.
http://www.artofproblemsolving.com/Foru … 7&t=607094

At first when I started to solve the equation of thought that you can specify only one factor.

Were you can ask any ratio

And the solution of the equation:

- any integer.

In order to attract attention to the method we have to solve a simple equation.

It is clear that in equation:

Decisions are determined by the solutions of the Pell equation:

We need to write a formula that clearly shows what was the substitution desired.  Will make a replacement.

Then the solutions are of the form:

It is necessary to consider another solution. In known solutions

- to find their counterparts. Upon substitution into the formula they give solutions. Are they using the formula.

We must be careful that the signs not to confuse them.

The Pell Equation: 

Is very simple. And for the first solution: 

  ; 

You can find the rest by the formula:

;

- any previous solution of the Pell equation.

As I said, for 8 unknown parameters and the formula goes bulky.

 

3 - the formula looks like this: http://math.stackexchange.com/questions … 527#738527 

Will consider here the special case when: 

 

 
Then the solutions are of the form: 

 

 

 

 

 

- integers asked us.  It is clear that if you will satisfy the condition

we can always write such a simple solution. It is easy enough to see how it turns out.

Forgot in the last formula, we need to 4 be reduced. -

Already corrected and reduced.

Equation:

Has a solution:

- any integer.

Equation:

Using the solutions of the Pell equation:   

Solutions have the form:

On Fildovsky award nominated    Manjul Bhargava.

http://www.mathunion.org/general/prizes/2014/

His work there.

http://arxiv.org/pdf/1006.1002v2.pdf

http://arxiv.org/pdf/1007.0052v1.pdf

Funny.  He can't solve a single equation.  Can't write a single formula.
Even says on the contrary that the formulas cannot be obtained.
But it's not.  In some cases, to obtain a formula for the solution.

They strongly opposed the formulas.

Do there exist four distinct integers such that the sum of any two of them is a perfect square?

This is equivalent to solving the following system of equations:

Let:

- any asked us integers.

For ease of calculation, let's make a replacement.

Then the solutions are of the form:

The system of Diophantine equations:

Solutions have the form:

An interesting case when:

  For this we need to solve the Pell equation: 

And solutions to substitute in the formula.

Equation: 

If the ratio is the square. 

Using the solutions of the equation Pell. 

Then the solutions are.