Skrytý text:Suppose
.
cases:
- If
, then
, so
, so
. Then we've got
, so
.
Then
and
or
and
or
and
or
,
, no solution.
- If
, then
, so
, so
. This case we've already solved.
- If
, then
, so
, so
. We've got
, or
.
For
we got solution only for
, then
. Suppose
.
If
, then
, so
and
, so
, contradiction. So
.
- If
, then
. For
or
it's not true. We prove
for
. For
it is true. We need to prove
, which is equivalent to
, which is true for
.
- If
, then
. For
it's not true. We prove
for
. For
it is true. We need to prove
, which is equivalent to
, which is true for
.
- If
, then
or
, no solution for
.
We got the only solution
.
Skrytý text:Suppose
. Then
, so
, so
, so
and
.
This implies
(then no solution) or
(then
, so no solution).
The equation has no solution.
Skrytý text:Suppose
. Then
, so
or
.
If
then
, impossible.
If
and we've got
. For
no solution. Let
. If
, then
, which is impossible. So
. Solution only for
, then
.
The solutions of equation are
Skrytý text:Suppose
. Then
and
.
Then
, so
. For
is
, which is impossible. So
.
If
, then
, which is impossible. So
or
.
If
we've got
, so again we must have
. We've got solution only for
. Then
.
If
we've got
. So
. If
then
, which is impossible. So
. We've got solution only for
,
, where
or
.
After permuting
we together have
solutions.