This equation was drawn to my attention with the instruction "solve for $n$."
$6n-\frac{n}{6}+n^6=n!-6^n$
Most of this is easy. Take the umpth root, multiply by the denominator, stuff like that. My only issue is finding a method to remove the factorial operator whilst keeping the other side of the equal sign satisfied.
Is there a quick method of doing this? Is there something easier than trying every possible combination and adjusting the solution to fit?
Any help would be appreciated.
-Thanks
On
As soon as $n!$ exceeds $6^n$, which is $n=12,$ the rest of the terms won't matter because they are too small. Actually the $6^n$ is big enough to matter up to $n=14$. That isn't many to try. Then $\frac n6$ is non-integer unless $n=0,6,12$ so you can only try those three. Only $0$ works. Done.
Hint : $n! >> 6^n > n^6 > n$ for large enough $n$