19. Determine the largest prime number less than 5000 of the form $a^{n}-1$, where $a$ and $n$ are positive integers, and $n$ is greater than 1 .
19. Answer: 127
Since $a^{n}-1=(a-1)\left(a^{n-1}+a^{n-2}+\cdots+a+1\right)$ is a prime, then $a=2$. Suppose $n=r s$ is a composite, then $2^{r s}-1=\left(2^{r}-1\right)\left(2^{r(s-1)}+2^{r(s-2)}+\cdots+2^{r}+1\right)$, where each factor on the right is greater than 1 , contradicting the fact that $a^{\prime \prime}-1$ is a prime. Therefore $n$ must be a prime. The largest prime $n$ such that $2^{n}-1<5000$ is 11 . However, $2^{11}-1=2047$ $=23 \times 89$, which is not a prime. Since $2^{7}-1=127$ is a prime number, the answer to this question is 127