Six, in all integers that start and end with 1 and alternate between 1 and 0 (i.e., $101$, $10101$, $1010101$, etc.), how many of them are prime numbers?
Six, Proof that there is only one prime number 101.
If $n \geqslant 2$, then $A=10^{2 n}+10^{2 n-2}+\cdots+10^{2}$
$$
+1=\frac{\left(10^{\mathrm{n}+1}-1\right)\left(10^{\mathrm{n}+1}+1\right)}{99} \text {. }
$$

When $n=2 m+1$, $\frac{10^{2 m+2}-1}{99}=10^{2 \mathrm{~m}}$ $+\cdots+10^{2}+1, A$ is a composite number;
When $n=2 m$, $9 \mid\left(10^{n+1}-1\right)$,
$11 \mid\left(10^{\mathrm{n}+1}+1\right), A$ is a composite number.