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II. Conditions of resolvability of the equation 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).
Sowhere- integers, 
Let's write down the solution of the equation:
Let's put, then:
 
If to take,
 
, then we will receive:
Let's inspect the decision:
:
Thus.
So, formulas give the solution of the equation.
In order that the equation had the decision in integers, it is necessary thatwere the whole positive numbers at integers.
Let
 
, where - the whole positive numbers 
 
From we will receive:
 
 , then.
 
From here we find, from equality we find.
 
So: , are criteria of resolvability of the equation in integers.
*u-can have an infinite number of integer solutions.
 

 If ,  are the whole positive numbers (-whole

 

 

 numbers), the equation разрешимо in integers.

 
Formulas of the solution of the equation can be written down in a look:
If the number   nonintegral, the solution of the equation  is ambiguous.
Consequence 1:
From follows, if  , and , the equation has the following decision:
Consequence 2:
From follows, if , and :
Consequence 3:
From 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:
 
а) Let, then .  Let , then.
 
So, if , the general multiplier of numbers is equal
 
.
 
If , that .
 
So, if , and , numbers  have the general multiplier
 
.
 
б) Let , then .  If , that.
 
Thus, if, and, the general multiplier of numbers is equal
 
.
 
Let, 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:
 
 
If, that, in this case the general multiplier of numbers is equal.
Consequence 4:
From follows, numbers surely have the general divider:
 
а), where  such that  was an integer;
 
б);
в) - if number , where - an integer;
г) - if number , where - an integer.
The general divider of numbers can be any natural number more unit.
The general divider of numbers can be both simple, and compound number.
At , - 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.
Let's show it:
.
Let, 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.