Math Is Fun Forum

Now will show you a different approach using the equations Pell.

For the equation:

Will set the number

- and can be of any sign.  Then use the Pell equation.

Where:

Then the solutions are.

And another solution:

Be aware that if the ratio of the Pell equation

- fold the square, can be reduced. In the formula, too, should be reduced.

In order to solve the equation:

Write the equation in General form.  Select the condition that the number of integer:

In order to record the decisions we make use of the solutions of the Pell equation of this type.

Numbers: 

- Ask us and can have any sign.  Knowing the solutions of the Pell equation we can write the solution of this equation as:

If

- not a whole. Find an equivalent replacement form 

on

  .  Then the equation becomes:

You will have to find out if the equation solutions.

This was easy to do because   

  - unknown.  You can of course choose another replacement.

This is a variation on Fermat's theorem.  The number of solutions is nite.  Will write a more General equation:

If 

- Will ask themselves.  Then the solution can be written;

- integers asked us.  You must consider that you can solve then reduce by common divisor.

Well here!
Today posted on his Blog 200 post.
http://www.artofproblemsolving.com/Foru … ?u=206450&

Each post is at least one formula!

Maybe it is better to solve such a system the system of Diophantine equations:

Then the solution can be written.

- integers asked us.

For the equation:

The solution can be written using the Pell equation:

To find them easily. Knowing what one solution can be found on the following formula.

To begin to

with

To begin to

with

Then the formula of the solutions can be written.

The upper sign according to the decision of the Pell equation for

.

Lower

.

   - these numbers can have any sign.

Sierpinski in his book for the equation:

Meant this formula.

No.
I showed the formula how to find them.
This formula is equivalent to the Pell equation.

Only then should   

  multiply by 3.

For the equation:

- integers.  You can write solutions:

It is easy to see that we can cut.  But you can do different.

Ask any number and place it on the multipliers in the following way.

Then the other numbers are.

- any integer asked us.

The system of equations:

Solutions have the form:

- integers asked us.

I think not to introduce additional equations, and directly solve the system of equations.

Using integer parameters

- Will make a replacement.

Then the solution can be written as:

To obtain relatively Prime solutions - after substitution should be reduced to common divisor.

This problem was there.
It reduced the formula - so give the link to the topic.
http://math.stackexchange.com/questions … 7x-37-dots

When the number of variable in the system of Diophantine equations a little.  The number of solutions of course means the formula of solutions it is impossible to give. I think we should write some beautiful system.  You can for example put the following task: when consecutive squares have a certain look.

For example: sequential squares representable as sums of squares.

Quite simply, the decision to write. Ask these numbers -

. That the following sum is a multiple of 4 or odd.

Lay on multipliers and find -

.  Solutions have the form:

You can consider another combination.

You also can specify the number of -

. And to factor.

Find the number -

. And we write the solutions of the equation as:

You can also write a formula connecting these numbers, but it is bulky. This approach is more simple.
The same standard approach can be used if you want to choose another polynomial form.

Probably will be more evident if we write this equation:

For this we need to use the solutions of the Pell equation:

Then the solutions are of the form:

***

And accordingly:

From the last formula can be seen as if not asked number $t,k$ always come to Diophantine equation 4th or more from a single variable. Or two variables. Depending on how you want to solve it.

As Tue proved such equations if you have solution then the number of solutions of course.

Integers which we ask.  Any.
We would like.
It's not my fault that I don't understand. Google translates so.