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

You are not logged in.

#326 2015-01-28 20:10:02

individ
Member
Registered: 2014-03-16
Posts: 316

Re: Formulas for the solution of Diophantine equations.

For the equation:

You can write the solution as:

Or different.

- Integers asked us.  Not a lot not in a convenient form the recorded decision, but it is. Go to positive numbers is not difficult.

Offline

#327 2015-01-29 04:00:49

individ
Member
Registered: 2014-03-16
Posts: 316

Re: Formulas for the solution of Diophantine equations.

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

To find solutions easily. Knowing the first solution

-

Knowing one solution, the following can be found by the formula.

Knowing any solution and using it, you can find the solution to this equation by the formula.

Or different formula.

Offline

#328 2015-01-31 00:40:28

individ
Member
Registered: 2014-03-16
Posts: 316

Re: Formulas for the solution of Diophantine equations.

For the system of equations:

Rewrite all the same this solution to compare.

- coefficients specified by the problem statement. I was interested in other solutions when solutions are not multiples.
Managed to get while this decision.

Last edited by individ (2015-01-31 00:43:07)

Offline

#329 2015-02-01 05:07:32

individ
Member
Registered: 2014-03-16
Posts: 316

Re: Formulas for the solution of Diophantine equations.

Equation if we write in the General form:

If in this equation there any equivalent to a quadratic form in which the root is an integer.

Then there are solutions. They can be written by making the replacement.

Then decisions can be recorded and they are as follows:

***

***

***

Offline

#330 2015-02-05 02:38:10

individ
Member
Registered: 2014-03-16
Posts: 316

Re: Formulas for the solution of Diophantine equations.

When he wrote the equation he meant probably that entry.

It turns out, this equation has a connection with the Pell equation:

For

it is necessary to use the first solution
. For
it is necessary to use the first solution
.  Knowing what the decision can be found on the following formula.

Using the solutions of the Pell equation can be found when there are solutions.

Will make a replacement.

Then the solution can be written:

- integers asked us. May be necessary, after all the calculations is to obtain a relatively simple solution, divided by the common divisor.

Offline

#331 2015-02-06 18:50:29

individ
Member
Registered: 2014-03-16
Posts: 316

Re: Formulas for the solution of Diophantine equations.

I thought as this task to generalize and use for any numbers.  It turned out that you can do without calculations.  For the system of equations:

Enough to factor the following number:

Using these numbers you can easily write the solution of this system of equations.

Offline

#332 2015-02-08 02:13:52

individ
Member
Registered: 2014-03-16
Posts: 316

Re: Formulas for the solution of Diophantine equations.

For the equation:

write the formula so that it was easier to go through. To facilitate calculations will make the replacement.

Then decisions can be recorded and they are.

Offline

#333 2015-02-09 17:28:33

individ
Member
Registered: 2014-03-16
Posts: 316

Re: Formulas for the solution of Diophantine equations.

For such equations:

- you can specify any, then decisions can be recorded.

Offline

#334 2015-02-10 05:03:22

individ
Member
Registered: 2014-03-16
Posts: 316

Re: Formulas for the solution of Diophantine equations.

For the equation:

If the number

is the problem any, and
is such as this:

Then the solution can be written:

- integers which we are set.

It's my decision.
How did you solve the other you can see there.
http://mathoverflow.net/questions/38354 … 180#196180

Offline

#335 2015-02-11 04:39:54

individ
Member
Registered: 2014-03-16
Posts: 316

Re: Formulas for the solution of Diophantine equations.

For the equation:

If you can represent numbers as:

This decision when the coefficients are related through the equation of Pell.

To simplify calculations we will make this change.

Then the solution can be written:

Last edited by individ (2015-02-11 04:41:53)

Offline

#336 2015-02-14 04:59:25

individ
Member
Registered: 2014-03-16
Posts: 316

Re: Formulas for the solution of Diophantine equations.

If you solve the system of equations:

When the standard approach solution and using a replacement.

Then the solution can be written as :

Offline

#337 2015-02-14 18:44:37

individ
Member
Registered: 2014-03-16
Posts: 316

Re: Formulas for the solution of Diophantine equations.

For the system of equations:

If you can decompose the coefficient multipliers as follows:

Their work squares:

Then decisions can be recorded.

You can add another simple option.  If the ratio can be written as:

Then decisions can be recorded.

Last edited by individ (2015-02-19 02:06:24)

Offline

#338 2015-02-17 04:17:53

individ
Member
Registered: 2014-03-16
Posts: 316

Re: Formulas for the solution of Diophantine equations.

One of the solutions to the equation. To get infinite amount of different.

If you can imagine.

Then decisions can be recorded.

If you can imagine.

Then decisions can be recorded.

Offline

#339 2015-02-23 22:41:17

individ
Member
Registered: 2014-03-16
Posts: 316

Re: Formulas for the solution of Diophantine equations.

This system of Diophantine equations was proposed.   Form APMO 2000. 2.

Has this type:

One easy solution can be written as:

It is better to write such a decision.

- any integer.

Last edited by individ (2015-02-26 02:03:14)

Offline

#340 2015-03-04 18:20:43

individ
Member
Registered: 2014-03-16
Posts: 316

Re: Formulas for the solution of Diophantine equations.

Equation;

Decisions are not always there. To find out, I use too much.  Decompose the multiplier number.

Finding the number \$t\$ and \$k\$, the solution writes:

Offline

#341 2015-04-05 03:30:31

Olinguito
Member
Registered: 2014-08-12
Posts: 649

Re: Formulas for the solution of Diophantine equations.

Hello individ. Hope you don't mind me contributing to your thread. I just considered diophantines of the form

This is solvable if and only if gcd(a,b) divides c. The general solution is

where

and X=x, Y=y is a particular solution. See this post:

Last edited by Olinguito (2015-04-05 03:49:20)

Bassaricyon neblina

Offline

#342 2015-04-08 18:12:46

individ
Member
Registered: 2014-03-16
Posts: 316

Re: Formulas for the solution of Diophantine equations.

This equation is a variation on this type of equations.
http://math.stackexchange.com/questions … e-patterns

If we consider the equation of a certain type.

The solutions can be written simply using the sequence.  The next element which is obtained from the previous one.

If we use the first element of the sequence.

Then the formula will look like.

If we use the first element of the sequence.

Then the formula will look like.

Offline

#343 2015-05-17 17:41:47

individ
Member
Registered: 2014-03-16
Posts: 316

Re: Formulas for the solution of Diophantine equations.

For generalization of the Pell equation question arises constantly. For example in this thread.

http://math.stackexchange.com/questions … 363#831363

So writes the Pell equation in General form.

If we know any solution of this equation.

If we use any solutions of the following equation Pell.

Then the following solution of the desired equation can be found by the formula.

We must remember that then can be reduced by a common factor.

Offline

#344 2015-06-09 20:35:24

individ
Member
Registered: 2014-03-16
Posts: 316

Re: Formulas for the solution of Diophantine equations.

The system of equations.

Equivalent to the need to solve the following system of equations.

To ease calculations we change.

Then we need the number to obtain Pythagorean triples can be found by the formula.

At any stage of the computation can be divided into common divisor.

Offline

#345 2015-06-20 00:13:00

individ
Member
Registered: 2014-03-16
Posts: 316

Re: Formulas for the solution of Diophantine equations.

For the system of equations.

If we do the replacement.

Then the solution will have the form:

Offline

#346 2015-07-14 20:11:56

individ
Member
Registered: 2014-03-16
Posts: 316

Re: Formulas for the solution of Diophantine equations.

For the system:

The solutions can be written as.

integers.

Offline

#347 2015-07-29 00:29:19

individ
Member
Registered: 2014-03-16
Posts: 316

Re: Formulas for the solution of Diophantine equations.

To solve the system of equations:

It is better to use such solutions.

Offline

#348 2015-08-04 18:22:36

individ
Member
Registered: 2014-03-16
Posts: 316

Re: Formulas for the solution of Diophantine equations.

One approach is to write the General formula.  For the equation.

The formula looks for simplicity without coprime solutions.

After substituting numbers

to reduce common divisor.

in such a number.

in such a number.

Offline

#349 2015-08-18 04:27:03

individ
Member
Registered: 2014-03-16
Posts: 316

Re: Formulas for the solution of Diophantine equations.

The system of book 2 tasks 22 , 23 .  The system is from the book of Diophantus.

Found this solution, but it sets a very different kind of decision. Different from the previous one.

*****************

Last edited by individ (2015-08-18 19:01:02)

Offline

#350 2015-08-19 01:07:30

individ
Member
Registered: 2014-03-16
Posts: 316

Re: Formulas for the solution of Diophantine equations.

For the system of book 2 task 30.

Decisions will be.

Offline