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:

]]>Has this type:

One easy solution can be written as:

- integers which we ask.It is better to write such a decision.

- any integer.]]>If you can imagine.

Then decisions can be recorded.

If you can imagine.

Then decisions can be recorded.

- integers which we ask.]]>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.

]]>When the standard approach solution and using a replacement.

Then the solution can be written as :

- integers which we ask.]]>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:

- integers which we ask.]]>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

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.

- integers, which we ask.]]>Enough to factor the following number:

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

]]>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.]]>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:

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***

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- integers asked us.]]>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. - any integer asked us.]]>

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.

]]>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.]]>