Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

You are not logged in.

- Topics: Active | Unanswered

**individ****Member**- Registered: 2014-03-16
- Posts: 289

Perhaps you should write a more General solution of the equation.

If you ask any integers:

- and use the solutions of the equation Pell.The solution, you can write:

You can write a simple solution without the Pell equation.

If:

If:

Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 289

In General formula generic for Pythagorean triples looks a little different.

If the number can be represented as a sum of squares.

The solution has the form:

Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 289

One said that this equation has no solutions. Because the equation Farm has no solutions. It turned out that the solution of this equation is.

Fermat's theorem cannot be proved, if you do not know how to solve Diophantine equations.

This approach cannot be used. This equation has a solution.

The solutions have the form:

- any integer.Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 289

You can write another solution. This is equivalent to the equation:

The solutions have the form:

................................................

..............................................

Will make a replacement.

The solution has the form:

Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 289

Then this equation. If there is a solution - they are infinitely many.

Finding solutions-it factor. To factor.

We use the solutions of the equation Pell.

Then the solutions are.

Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 289

Equation:

You can write a simple solution:

- any integer asked us.Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 289

Equation:

If the ratio is the square.

Using the solutions of the equation Pell.

Then the solutions are.

Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 289

In the equation.

Solutions are provided by the Pell equation.

And have a look.

Solving the Pell equation can be found. Knowing the past can be found .

You can start with.

All numbers can have any sign.

Another can be reduced to 4. And come to the equation.

Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 289

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.

Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 289

For the system of equations:

- choose an integer, so that the bracket was intact.Then the solutions are.

Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 289

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:

Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 289

This system was solved and there.

http://www.artofproblemsolving.com/Foru … 6&t=602478

You can compare solutions.

Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 289

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.

Offline

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 107,167

Fildovsky award? What is that?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 289

There you can see.

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

As not fair. He no single formula did not write and say it is good work.

I wrote so many formulas and are not allowed to publish.

Offline

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 107,167

He is a Fields medal winner.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 289

Equation:

Using the solutions of the Pell equation:

Solutions have the form:

Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 289

Equation:

Has a solution:

- any integer.Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 289

Equation:

Has a solution:

- any integer.

*Last edited by individ (2014-08-24 02:52:05)*

Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 289

Equation:

Decisions will be :

...............

................

.................

If you use the solutions of the Pell equation:

And the following substitutions:

Or:

Then the solutions are of the form:

*Last edited by individ (2014-09-01 20:58:33)*

Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 289

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

Already corrected and reduced.

*Last edited by individ (2014-09-01 21:00:00)*

Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 289

You can write such a formula.

Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 289

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.

Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 289

It is necessary to say a few words about the formula about which I have spoken. For the equation:

Need to write this simple formula.

I think that this formula gives all solutions. Mutually simple solution obtained after reduction to common divisor.

For example there was a similar situation with the equation:

It is enough to write the formula generates an endless series of decisions in all degrees. For this we use the Pythagorean triple.

And the number of their sets.

- what some integers. Then the solution can be written.And mutually simple solutions can get if you cut down on common divisor. Although there will be not one simple solution.

Offline

**individ****Member**- Registered: 2014-03-16
- Posts: 289

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.Offline