Integer solutions for the equation $2r = q^{r-1}.$

Question

I want to find all the integer solutions for the following equation $2r = q^{r - 1}$ for $r\geq 1$ and $q$ an integer i.e., $q=2k$ for some natural number k not equal to zero.

I know that we should the ln function on both sides of the equation but how I will get the only solutions $q=4$ when $r=2$ and $q=2$ when $r=4$.

Could someone explain this to me please?

Answer 1

If $2r = q^{r - 1}$ for some natural numbers $r\ge1$ and $q$ then $r\ge2$, $q=2k$ for some integer $k\ge1$, and $$r=2^{r-2}k^{r-1}.$$

• For $r=2$, the solution is $k=2$, i.e. $q=4$.
• For $r=3$, there is no solution.
• For $r=4$, the solution is $k=1$, i.e. $q=2$.
• For $r\ge5$, there is no solution because $r<2^{r-2}$.

Answer 2

Clearly $q>1$ and so $r=\frac{q^{r-1}}{2}\geq2^{r-2}$, which implies $r\leq4$.