13. (Full score 12 points) How many prime numbers are there among all integers that start and end with 1 and alternate between 1 and 0 (such as 101, 10101, 1010101, ...)? Why? And find all the prime numbers.
13. The following proves that $10101, 1010101, \cdots \cdots$, are not prime numbers.
For $n \geqslant 2$, then
$$
\begin{aligned}
A & =\underbrace{1010101 \cdots 01}_{2 n} \\
& =10^{2 n}+10^{2 n} 2+\cdots+10^{2}+1 . \\
& =\frac{10^{2 n+2}-1}{10^{2}-1}=\frac{\left(10^{n+1}+1\right)\left(10^{n+1}-1\right)}{99} .
\end{aligned}
$$

If $n$ is odd, i.e., $n=2 m+1$, then
$$
\frac{10^{n+1}-1}{99}=\frac{10^{2 n+2}-1}{99}=\underbrace{1010 \cdots 101}_{2 m} \text {. }
$$

Thus, $A=\underbrace{10101 \cdots 101}_{2 m \text { terms }} \times\left(10^{n+1}+1\right)$.
Hence, $A$ is a composite number.
If $n$ is even, then $9\left|\left(10^{n+1}-1\right), 11\right|\left(10^{n+1}+1\right)$, so $A$ is also a composite number.
Since 101 is a prime number, the only prime number in the sequence is 101.