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.