II. Conditions of resolvability of the equation
in integers.
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image1.gif)
We solve the equation
using a new method of the solution of the equations of Eremin (see M.A.Eremin “A new method of the solution of the equations” Arzamas 2000).
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image2.gif)
So
, where![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image4.gif)
,
- integers, ![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image155.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image3.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image4.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image5.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image6.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image155.gif)
Let's write down the solution of the equation
:
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image8.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image156.gif)
Let's put
, then:
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image11.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image12.gif)
If to take
,
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image13.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image14.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image157.gif)
Let's inspect the decision:
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image19.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image158.gif)
Thus
.
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image21.gif)
So, formulas
give the solution of the equation
.
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image22.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image23.gif)
In order that the equation
had the decision in integers, it is necessary that
,
were the whole positive numbers at
integers.
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image24.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image25.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image26.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image27.gif)
Let
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image28.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image29.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image30.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image31.gif)
From
we will receive:
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image32.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image33.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image34.gif)
From here we find
, from equality
we find
.
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image35.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image36.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image37.gif)
So:
, ![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image39.gif)
are criteria of resolvability of the equation
in integers.
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image38.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image39.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image40.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image41.gif)
*u-can have an infinite number of integer solutions.
If
numbers), the equation |
![]() |
Formulas
of the solution of the equation
can be written down in a look:
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image48.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image49.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image159.gif)
If the number
nonintegral, the solution of the equation
is ambiguous.
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image54.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image55.gif)
Consequence 1:
From
follows, if
, and
, the equation
has the following decision:
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image56.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image57.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image58.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image59.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image160.gif)
Consequence 2:
From
follows, if
, and ![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image65.gif)
:
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image63.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image64.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image65.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image66.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image161.gif)
Consequence 3:
From
follows, numbers
surely have the general multiplier.
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image70.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image71.gif)
The general multiplier can accept the following values:
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image162.gif)
Let's show it.
Numbers
for the equation
are equal:
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image75.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image76.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image163.gif)
а) Let
, then
. Let ![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image82.gif)
, then
.
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image80.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image81.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image82.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image83.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image84.gif)
So, if ![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image85.gif)
, the general multiplier of numbers
is equal
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image85.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image86.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image87.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image88.gif)
If ![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image89.gif)
, that ![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image91.gif)
.
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image89.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image90.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image91.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image92.gif)
So, if
, and
, numbers
have the general multiplier
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image93.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image94.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image95.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image96.gif)
б) Let
, then
. If
, that![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image100.gif)
.
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image97.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image98.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image99.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image100.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image101.gif)
Thus, if
, and
, the general multiplier of numbers
is equal
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image102.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image103.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image104.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image105.gif)
Let![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image106.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image107.gif)
, then, or
. In this case general multiplier of numbers
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image106.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image107.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image108.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image109.gif)
it is equal
.
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image110.gif)
в) Let
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image113.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image111.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image112.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image113.gif)
If![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image114.gif)
. So, if
, and
, the general multiplier of numbers
is equal:
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image114.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image115.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image116.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image117.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image118.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image119.gif)
If
, that
, in this case the general multiplier of numbers
is equal
.
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image120.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image121.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image122.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image123.gif)
Consequence 4:
From
follows, numbers
surely have the general divider:
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image124.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image125.gif)
а)
, where
such that
was an integer;
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image126.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image127.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image128.gif)
б)
;
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image129.gif)
в)
- if number
, where
- an integer;
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image130.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image131.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image132.gif)
г)
- if number
, where
- an integer.
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image133.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image134.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image135.gif)
The general divider of numbers
can be any natural number more unit.
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image136.gif)
The general divider of numbers
can be both simple, and compound number.
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image136.gif)
At
,
- simple number, minimum possible.
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image137.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image138.gif)
The equation
if
does not allow
to make in numbers reduction on their general divider. At reduction the equation essence is broken.
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image139.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image140.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image141.gif)
Let's show it:
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image142.gif)
Let
, where
- the whole positive numbers.
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image143.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image144.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image145.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image146.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image147.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image148.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image149.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image150.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image151.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image152.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image153.gif)
![](http://mathematicasu.406.com1.ru/images/formula/glv2/Image154.gif)