III. Examples of the solution of the equations of a look 
.
Example 1. To find the solution of the equation 
in integers.
Decision:
We define, разрешимо this equation in integers or not. We calculate criteria of resolvability of the equation:
at 
.
From 
, 
it is visible at 
number 
are whole positive therefore this equation has the decision in integers.
On the general formulas 
we will write down the decision:
Example 2.
To find the solution of the equation 
in integers.
Decision:
We define, whether разрешимо the equation in integers. On formulas 
at 
we find:
Let 
=6834, then 
.
So, numbers 
whole positive, this equation разрешимо in integers.
On the general formulas 
we find the solution of the equation 
:
Example 3.
To find the solution of the equation 
.
Decision:
We define, whether разрешимо this equation in integers.
On formulas 
at 
we find:
.
If 
that 
.
As 
,
- the whole positive numbers at 
that this equation разрешимо in integers.
Let's write down on the general formulas 
the solution of this equation:
Example 4.
To find the decision in equation integers 
.
Decision:
We define, whether has this equation the decision in integers. We calculate criteria of resolvability of the equation:
at 
If 
, 
Numbers 
- whole positive, this equation разрешимо in integers.
On the general formulas 
we will write down the solution of the equation 
:
Example 5.
To find the solution of the equation 
in integers.
Decision:
As 
that we apply a consequence No. 2:
This equation разрешимо in integers. Here solution of this equation:
If to calculate criteria of resolvability of the equation, we will receive
,
If 
.
So, 
Thus also we will receive the decision 
.
Example 6.
To find the solution of the equation 
in integers.
Decision:
We calculate criteria of resolvability of the equation 
,
at 
At 
, we will receive 
,
As 
, - the decision in integers has the whole positive numbers, this equation.
On the general formulas 
we will write down the solution of the equation:
Example 7.
To find the solution of the equation 
in integers. To show as the method of infinite descent operates for this example.
Decision:
We calculate for this equation criteria of resolvability: 
at 
If 
, we will receive: 
As 
- the decision in integers has the whole positive numbers, this equation.
Let's write down on the general formulas the decision:
From 
it is visible, if 
, where 
, 
- integers. Thus, we will receive infinite sequence of numbers 
, that is degrees in numbers 
will change 
:
Let's tabulate data on:
 |
6 | 21 | 36 | 51 | 66 | 81 | 96 | 111 | 126 | 141 | 156 | 171 |
 |
5 | 17 | 29 | 41 | 53 | 65 | 77 | 89 | 101 | 113 | 125 | 137 |
 |
8 | 28 | 48 | 68 | 88 | 108 | 128 | 148 | 168 | 188 | 208 | 228 |
 |
186 | 201 | 216 | 231 | 246 | 261 | 276 | 291 | 306 | 321 | 336 |
 |
149 | 161 | 173 | 185 | 197 | 209 | 221 | 233 | 245 | 257 | 269 |
 |
248 | 268 | 288 | 308 | 328 | 348 | 368 | 388 | 408 | 428 | 448 … |
Thus, we have shown a so-called method of infinite descent on an equation example 
Example 8.
To find the decision in equation integers 
Decision:
We calculate criteria of resolvability of this equation in integers: 
at 
.
Let 
, 
As numbers 
- whole positive, this equation разрешимо in integers. On the general formulas we will write down the decision:
Example 9.
To define, whether has the equation 
the decision in integers. If yes, that to find these decisions.
Decision:
We calculate criteria of resolvability of the equation at 
:
If 
, 
As numbers 
- whole positive, this equation has the decision in integers.
On the general formulas we will write down this decision:
Example 10.
To find the decision in equation integers 
.
Decision:
We calculate criteria of resolvability of the equations in integers at 
:
.
If 
that: 
.
As numbers 
- whole positive, this equation has the decision in integers.
Let's write down this decision:
Example 11.
To solve the equation 
in integers.
Decision:
We define criteria of resolvability of the equation in integers: 
at 
.
If 
, 
As numbers 
- whole positive, this equation разрешимо in integers:
Example 12.
To solve the equation 
in integers.
Decision:
We calculate criteria of resolvability of this equation at 
:
As at 
- even, expression 
- odd number, 
- nonintegral number. In this case the solution of the equation is ambiguous. It can have or not have decisions in integers. Let's check it. Let's copy the equation thus: 
. In the following heads we will consider solutions of the equations 
in integers. And here we will simply write down the decision the equation 
:
Or:
Example 13.
To define, whether has the decision the equation 
in integers.
Decision:
We define criteria of resolvability of this equation in integers at 
:
Let 
, then 
As numbers 
- whole positive, this equation has the decision in integers. Let's write down the decision:
From 
it is visible, numbers 
can be whole:
1) if 
;
2) if 
- rational, for example:
where 
- integers.
From 
it is visible, numbers 
can be nonintegral if 
integers there is more than unit.
Example 14.
To define, whether has the decision in integers the equation 
.
Decision:
We calculate criteria of resolvability of the equation in integers 
,
at 
:
,
If 
, 
Numbers 
- whole, this equation has the decision in rational numbers, as 
. Let's write down the solution of the equation:
Example 15.
To define, whether has the decision in integers the equation 
.
Decision:
We define criteria of resolvability of the equation in integers:
If 
, 
Let's write down the solution of the equation:
From 
it is visible, numbers can be nonintegral, as 
If to take 
, we will receive:
The solution of the equation we will write down in a look:
Numbers also nonintegral.