Find all the positive integers such that .
I could show that no solution is there if .
But the remaining cases still bug me.
any help.
Write (a) . Clearly, and so . Note that since is a fourth power, then must be an integer.*
It can be shown# that for any integer , so implies . Now, taking we have .
The only possible value for are and . The corresponding equations for (a) are , , , and . Only the first two are solvable. Therefore, the only solutions to the originial equation are those that Also sprach Zarathustra and abhishekkgp gave.
*Not necessary for the solution, only to reduce the number cases.
#If , then . Adding gives and . Using we have .
That was not a typo but a serious mistake. We have .
For , the inequality can be shown by induction on . We can check that . Assuming that multiply both sides by . Then . Now it's left to show that . Note that .
Since gets smaller as gets larger it follows that and therefore - this completes the inductive part.
Since we want , we must take . Otherwise and the inequality holds.