II. Conditions of resolvability of the equation
in integers.
in integers.
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).
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).
So
, where
, 
- integers, 
, where, - integers,
Let's write down the solution of the equation
:
:
Let's put
, then:
, then:
If to take
,
,
, then we will receive:
Let's inspect the decision:
:
Thus
.
.
So, formulas
give the solution of the equation
.
give the solution of the equation.
In order that the equation 
had the decision in integers, it is necessary that
, 
were the whole positive numbers at 
integers.
had the decision in integers, it is necessary that, were the whole positive numbers at integers.
Let
, 
, where 
- the whole positive numbers 
From
we will receive:
we will receive:
, then
.
From here we find
, from equality
we find
.
, from equality we find.
So: 
, 
are criteria of resolvability of the equation 
in integers.
, are criteria of resolvability of the equation in integers.
*u-can have an infinite number of integer solutions.
|
If
numbers), the equation |
 |
, 
are the whole positive numbers (
-whole
Formulas
of the solution of the equation 
can be written down in a look:
of the solution of the equation can be written down in a look:
If the number 
nonintegral, the solution of the equation 
is ambiguous.
nonintegral, the solution of the equation is ambiguous.
Consequence 1:
From
follows, if 
, and 
, the equation 
has the following decision:
follows, if , and , the equation has the following decision:
Consequence 2:
From
follows, if 
, and 
:
follows, if , and :
Consequence 3:
From
follows, numbers
surely have the general multiplier.
follows, numbers surely have the general multiplier.
The general multiplier can accept the following values:
Let's show it.
Numbers 
for the equation 
are equal:
for the equation are equal:
а) Let
, then 
. Let 
, then
.
, then . Let , then.
So, if 
, the general multiplier of numbers
is equal
, the general multiplier of numbers is equal 
.
If 
, that 
.
, that .
So, if 
, and 
, numbers 
have the general multiplier
, and , numbers have the general multiplier 
.
б) Let 
, then 
. If 
, that
.
, then . If , that.
Thus, if
, and
, the general multiplier of numbers
is equal
, and, the general multiplier of numbers is equal
.
Let
, then, or
. In this case general multiplier of numbers
, then, or. In this case general multiplier of numbers
it is equal 
.
.
в) Let 
If
. So, if
, and
, the general multiplier of numbers
is equal:
. So, if, and, the general multiplier of numbers is equal:
If
, that
, in this case the general multiplier of numbers
is equal
.
, that, in this case the general multiplier of numbers is equal.
Consequence 4:
From
follows, numbers
surely have the general divider:
follows, numbers surely have the general divider:
а)
, where 
such that 
was an integer;
, where such that was an integer;
б)
;
;
в)
- if number 
, where
- an integer;
- if number , where - an integer;
г)
- if number 
, where
- an integer.
- if number , where - an integer.
The general divider of numbers
can be any natural number more unit.
can be any natural number more unit.
The general divider of numbers can be both simple, and compound number.
can be both simple, and compound number.
At 
,
- simple number, minimum possible.
, - simple number, minimum possible.
The equation 
if 
does not allow
to make in numbers reduction on their general divider. At reduction the equation essence is broken.
if does not allow to make in numbers reduction on their general divider. At reduction the equation essence is broken.
Let's show it:
.
Let
, where
- the whole positive numbers.
, where - the whole positive numbers.
or:
. If to make division of numbers
into their general divider
, we will receive:
or:
. Let's receive the equation not equivalent to the equation
. Reduction of numbers on their general divider possibly only for the equation
:
, or
- the essence of the equation has not changed.