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

You are not logged in.

#126 2014-05-03 00:26:08

individ
Member
Registered: 2014-03-16
Posts: 302

Re: Formulas for the solution of Diophantine equations.

Okay. I think it is necessary to show the formula, in order to more constructive discussion went.
The system:

Solutions can be written as:

Or this:

Offline

#127 2014-05-03 23:12:50

individ
Member
Registered: 2014-03-16
Posts: 302

Re: Formulas for the solution of Diophantine equations.

The system of equations with given coefficients:

has the solutions:

Offline

#128 2014-05-05 18:32:59

individ
Member
Registered: 2014-03-16
Posts: 302

Re: Formulas for the solution of Diophantine equations.

In the system of equations:

Another solution can be written.

All three formulas derived me just describe all solutions of the system. I think the question can be considered closed.

Offline

#129 2014-05-06 18:03:26

individ
Member
Registered: 2014-03-16
Posts: 302

Re: Formulas for the solution of Diophantine equations.

Solving the equation.

got some solutions, but still the question remains. Below are all the decisions or not?

And more.

Last edited by individ (2014-05-06 21:54:06)

Offline

#130 2014-05-07 16:50:22

individ
Member
Registered: 2014-03-16
Posts: 302

Re: Formulas for the solution of Diophantine equations.

It's pretty old equations that are solved by Euler.
the equation:

If we use the solutions of Pell's equation:

Solutions can be written:

- We ask ourselves.  While the formula and can be written differently.

Offline

#131 2014-05-13 05:35:42

individ
Member
Registered: 2014-03-16
Posts: 302

Re: Formulas for the solution of Diophantine equations.

For the equation:

Write down the solution when the number can be factored as follows.

Then use the solution of Pell's equation:

Where the coefficient is given by:

Then the solution can be written:

And more.

Offline

#132 2014-05-13 17:34:49

individ
Member
Registered: 2014-03-16
Posts: 302

Re: Formulas for the solution of Diophantine equations.

Probably it is necessary to draw a formula for the solution in the general form:

In the equation:

Solutions can be written:

- what some integers.

Offline

#133 2014-05-14 00:45:24

individ
Member
Registered: 2014-03-16
Posts: 302

Re: Formulas for the solution of Diophantine equations.

Sometimes you have to deal with this equation:

- integer coefficients.

I wrote below - to start a particular solution of Diophantine equations.

To do this, use the solutions of Pell's equation:

I turned solutions such.

And more:

Offline

#134 2014-05-15 18:04:29

individ
Member
Registered: 2014-03-16
Posts: 302

the equation:

Has a solution:

Has a solution:

Offline

#135 2014-05-16 01:22:05

individ
Member
Registered: 2014-03-16
Posts: 302

Re: Formulas for the solution of Diophantine equations.

I solved the equation

It turned out one interesting detail. This equation always has solutions in integers.
I question went to the site and opened the topic. http://mathoverflow.net/questions/16632 … -equations
These idiots took the theme and blocked. Do not understand what to do what they do not understand all removed.

Offline

#136 2014-05-16 01:29:41

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Formulas for the solution of Diophantine equations.

Did you call them idiots when you posted your solution? That would get your stuff blocked.

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

#137 2014-05-16 01:37:22

individ
Member
Registered: 2014-03-16
Posts: 302

Re: Formulas for the solution of Diophantine equations.

It is idiots!
Now the formula potter. Check it.
And it turned out pretty interesting consequence that although the equation

may not always be solutions
But the equation

can always be written solutions in integers.
And they were not even allowed to discuss it!
They do not understand all the wash!

Offline

#138 2014-05-16 01:46:22

ShivamS
Member
Registered: 2011-02-07
Posts: 3,648

Re: Formulas for the solution of Diophantine equations.

Yes, they are sometimes harsh. Not all forums are as nice as here.

Offline

#139 2014-05-16 01:49:54

individ
Member
Registered: 2014-03-16
Posts: 302

Re: Formulas for the solution of Diophantine equations.

I discovered this by accident.
Quite an interesting event. Very few people could imagine that this is possible.
Instead of all let's understand washable.

Offline

#140 2014-05-16 03:10:22

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Formulas for the solution of Diophantine equations.

SO is for Topology, why are you posting in there? Also, your question is not readable. Go back to SE.

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

#141 2014-05-16 03:11:59

ShivamS
Member
Registered: 2011-02-07
Posts: 3,648

Re: Formulas for the solution of Diophantine equations.

Math Overflow is not only for topology?

Offline

#142 2014-05-16 03:19:54

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Formulas for the solution of Diophantine equations.

Seems more for pure and professional people, less for an amateur. He has had some success in SE, he will not have any in SO.

Pardon, it is not only about Topology.

Its density of expertise is currently skewed significantly towards number theory, algebraic and arithmetic geometry, category (and higher category) theory, and logic and set theory.

But to be fair the most likely reason for the 5 downvotes is that the question is unreadable as posted.

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

#143 2014-05-16 18:46:49

individ
Member
Registered: 2014-03-16
Posts: 302

Re: Formulas for the solution of Diophantine equations.

The interesting thing is that the equation:

If we use the equation Pell:

Coefficient is defined as follows:

- integers asked us.  Then the solutions are of the form:

And another formula.

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

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

In the second formula should be chosen so Pell's equation and its solution so that the fraction decreased and turned to an integer.

Offline

#144 2014-05-23 18:08:44

individ
Member
Registered: 2014-03-16
Posts: 302

Re: Formulas for the solution of Diophantine equations.

Solutions of the equation:

Can be written without using Pell's equation:

- what some integer.

Offline

#145 2014-05-26 18:19:19

individ
Member
Registered: 2014-03-16
Posts: 302

Re: Formulas for the solution of Diophantine equations.

It was necessary to prove the existence of solutions for all
Can all make it easier to prove that 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.

So there is always a solution.

Last edited by individ (2014-05-26 18:21:33)

Offline

#146 2014-05-27 04:22:41

individ
Member
Registered: 2014-03-16
Posts: 302

Re: Formulas for the solution of Diophantine equations.

Interested in the solution in general Diophantine equations of the form:

- what some integer.

Solutions similar equations can be written.

Since this equation is easy, as it is quite symmetrical.

Such a solution can write.

And solutions can be written:

Whether there are any thoughts how to solve this equation?

At first I thought to use for solving Pell's equation, but I think that you can do without.

Offline

#147 2014-05-27 19:08:41

individ
Member
Registered: 2014-03-16
Posts: 302

Re: Formulas for the solution of Diophantine equations.

Strangely enough, the solution is finite.

for the equation:

If it is possible to decompose the coefficient as follows:

Then the solutions are of the form:

Thought the solution is determined by the equation Pell, but when calculating the sign was a mistake. There's no difference, but the amount should be. Therefore, the number of solutions of course.

Offline

#148 2014-05-27 23:14:02

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Formulas for the solution of Diophantine equations.

Hi;

Post #144 can be shortened a bit

Nice formula by the way!

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

#149 2014-05-28 01:05:58

individ
Member
Registered: 2014-03-16
Posts: 302

Re: Formulas for the solution of Diophantine equations.

Solutions Pythagorean triples:

You can also submit through another Pythagorean triples:

Here is some formula, although they can write an infinite amount. All a matter of taste.

Or these:

Then using these triples can come to others.

- what some integers.

For one, this entire kindergarten. Why is he interested in threes.

Offline

#150 2014-05-29 17:19:46

individ
Member
Registered: 2014-03-16
Posts: 302

Re: Formulas for the solution of Diophantine equations.

I think that it is necessary to adjust the formula solutions.

For the equation:

Solutions can be written.

For the system of equations:

Solutions have the form:

- integers of any sign.

Offline